Hypergeometric special solutions for %The Pad\'e interpolation method applied to %additive difference $d$-Painlev\'e equations % (additive grid version)
Hidehito Nagao
We have studied Pad\'e interpolation problems on an additive grid, related to additive difference ($d$-) Painlev\'e equations of type $E_7^{(1)$, $E_6^{(1)$, $D_4^{(1)$ and $A_3^{(1)$. By solving those problems, we can derive time evolution equations, Lax pairs of scalar type and determinant formulae of special solutions, for the corresponding $d$-Painlev\'e equations. We show how the special solutions are expressed as determinant formulae by using terminating (generalized) hypergeometric functions. %, as far as we know.
1706.10101.tex
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\title{Hypergeometric special solutions for
%The Pad\'e interpolation method applied to 
%additive difference 
$d$-Painlev\'e equations
% (additive grid version)
}
\author{Hidehito Nagao}
\address{Department of Arts and Science, National Institute of Technology, Akashi College, Hyogo 674-8501, Japan}
\email{nagao@akashi.ac.jp}
\keywords{Pad\'e method, Pad\'e interpolation, additive difference Painlev\'e equation, hypergeometric function.}
\subjclass[2010]{33C05, 33C15, 33C20, 33D15, 33K20, 39A10, 34M55, 39A10, 41A05, 41A20}

\begin{document}

\begin{abstract}
We have studied Pad\'e interpolation problems on an additive grid, related to additive difference ($d$-) Painlev\'e equations of type $E_7^{(1)}$, $E_6^{(1)}$, $D_4^{(1)}$ and $A_3^{(1)}$. By solving those problems, we can derive time evolution equations, Lax pairs of scalar type and determinant formulae of special solutions, for the corresponding $d$-Painlev\'e equations. We show how the special solutions are expressed as determinant formulae by using terminating (generalized) hypergeometric functions.
%, as far as we know.
\end{abstract}

\maketitle
\tableofcontents
\renewcommand\baselinestretch{1.2}%行間調整

%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}\label{sec:intro}
\noindent
Firstly, in Sections \ref{subsec:Peq}, \ref{subsec:Lax} and \ref{subsec:sol} we introduce the background of time evolutions, Lax forms and hypergeometric solutions for the additive difference ($d$-) Painlev\'e equation. Furthermore in these subsections we mention only main results (i.e. Propositions \ref{prop:evolution}, \ref{prop:Lax}, and Theorem \ref{thm:sol}), for $d$-Painlev\'e equations of type $E_7^{(1)}$, $E_6^{(1)}$, $D_4^{(1)}$ and $A_3^{(1)}$, which are derived by Pad\'e interpolation method in Section \ref{sec:interpolation method}. Secondly, in Section \ref{subsec:pade method} we introduce the background of the Pad\'e method. In Section \ref{subsec:purpose} we state the purpose and the organization of this paper.
 
\subsection{The background of continuous/discrete Painlev\'e equations}\label{subsec:Peq}　\\

\noindent
Both the second order continuous and discrete Painlev\'e equation has been well studied in mathematics and physics (e.g. \cite{Conte99, Forrester10}). In the geometric approach, for each Painlev\'e equation, K. Okamoto constructed certain rational surfaces, called the ``{\it spaces of initial values}", which parametrize all the solutions \cite{Okamoto79}. Furthermore K. Takano found that the Painlev\'e equations are uniquely determined by the spaces of initial values \cite{MMT99, ST97}. Extending these works, H. Sakai 
 proposed a certain class of second order continuous/discrete Painlev\'e equations, as mentioned below.

In Sakai's theory \cite{Sakai01}, the continuou/discrete Painlev\'e equations have been classified on the basis of the spaces of initial values connected to extended affine Weyl groups. The spaces of initial values are obtained from $\P^2$ (resp. $\P^1 \times \P^1$) by blowing up at 9 (resp. 8) singular points. In view of the configuration of 9 (resp. 8) singular points in $\P^2$ (resp. $\P^1 \times \P^1$), there exist three types of discrete Painlev\'e equations and six continuous Painlev\'e equations in the classification: elliptic difference ($e$-), multiplicative difference ($q$-), additive difference ($d$-) and continuous (differential). The only $e$-Painlev\'e equation \cite{ORG01}, possessing the extended affine Weyl group symmetry of type $E_8^{(1)}$, is obtained from the most generic configuration. All of the other  Painlev\'e equations are obtained as its degeneration. 
 
The continuous/discrete Painlev\'e equations are classified into 22 cases below according to the following configuration.

{\arraycolsep=0.1pt
%\[
\begin{equation}\label{eq:classification}\nonumber
\begin{array}{ccccccccccccccccccc}
{\rm ell.\mbox{($e$-)}}\quad&E^{(1)}_8\\[-2mm]
&&&&&&&&&&&&&&&{\mathbb Z}\\[-2mm]
&\downarrow&&&&&&&&&&&&&\nearrow&&\searrow\\
{\rm mul.}\mbox{($q$-)}\quad&E^{(1)}_8&\rightarrow&E^{(1)}_7&\rightarrow&E^{(1)}_6
&\rightarrow &
%\underset{(\mbox{$q$-$P_{\rm VI}$})}{
D_5^{(1)}
%}
&\rightarrow&
%\underset{(\mbox{$q$-$P_{\rm V}$})}{
A_4^{(1)}
%}
&\rightarrow&
%\underset{(\mbox{$q$-$P_{\rm IV}$,$q$-$P_{\rm III}$})}{
(A_2+A_1)^{(1)}
%}
&\rightarrow&
%\underset{(\mbox{$q$-$P_{\rm II}$})}{
(A_1+A_1^{\prime})^{(1)}
%}
&\rightarrow&
%\underset{(\mbox{$q$-$P_{\rm I}$})}{
A_1^{(1)}
%}
&&{\mathcal D}_6\\[2mm]
&\downarrow&&\downarrow&&\downarrow&&&\searrow&&\searrow&&\searrow&&\searrow&&\searrow&\\
{\rm add.}\mbox{($d$-)}\quad
&E^{(1)}_8&\rightarrow&E^{(1)}_7&\rightarrow&E^{(1)}_6
&&\rightarrow&&\underset{(\mbox{$P_{\rm VI}$})}{D_4^{(1)}}&\rightarrow&\underset{(\mbox{$P_{\rm V}$})}{A_3^{(1)}}&\searrow
\rightarrow
&\underset{(\mbox{$P_{\rm III}$})}{(A_1+A_1^{\prime})^{(1)}}&\rightarrow&A^{(1)}_1\rightarrow&&{{\mathbb Z}_2}\\
&&&&&&&&&&&&\searrow&&\searrow&&&\downarrow\\
&&&&&&&&&&&&&\underset{(\mbox{$P_{\rm IV}$})}{A_2^{(1)}}&\rightarrow&\underset{(\mbox{$P_{\rm II}$})}{A_1^{(1)}}\rightarrow&&\underset{(\mbox{$P_{\rm I}$})}{1}\\
\end{array}
%\]
\end{equation}
}
\noindent
Here $A \to B$ represents that $B$ is obtained from $A$ by degeneration limit. We also  remark that the $d$-Painlev\'e equations of type $D_4^{(1)}$ and its degeneration ($A_3^{(1)}$, \ldots) arise as B\"acklund (Schlesinger) transformations of
differential Painlev\'e equations, namely $P_{\rm J}$ (${\rm J}={\rm I}$, ${\rm II}$, ${\rm III}$, ${\rm IV}$, ${\rm V}$ and ${\rm VI}$). 

Recently K. Kajiwara, M. Noumi and Y. Yamada have given a comprehensive review on the current status of achievements in
the geometric aspects of the Painlev\'e equations \cite{KNY17}.
 
In this paper we put an emphasis on studying the $d$-Painlev\'e equations of type $E_7^{(1)}$, $E_6^{(1)}$, $D_4^{(1)}$ and $A_3^{(1)}$ through Pad\'e interpolated problems. In Sections \ref{subsec:Lax} and \ref{subsec:sol} we briefly mention the background of Lax forms and hypergeometric special solutions for the $d$-Painlev\'e equations.  

\subsection{The background of Lax forms for $d$-Painlev\'e equations}\label{subsec:Lax}　\\

\noindent
It is well known that the continuous Painlev\'e equations are obtained from deformation theory of linear differential equations of $2 \times 2$ matrix type (see \cite{JM81-2}). The continuous Painlev\'e equations are given as the form of compatibility condition between a linear differential equation and its differential deformation equation. 

The $d$-Painlev\'e equations of type $D_4^{(1)}$ and its degeneration can be characterized by the same linear differential equation as the continuous Painlev\'e equations and its additive deformation (Schlesinger transformation). For example, some $2 \times 2$ matrix Lax pairs for types from $d$-$D_4^{(1)}$ to $d$-$A_1^{(1)}$ have been derived in \cite{GORS98}, using a Schlesinger transformation of differential equations. We call the linear differential equation ``{\it the differential Lax form}". The $d$-Painlev\'e equations of type $E_8^{(1)}$, $E_7^{(1)}$ and $E_6^{(1)}$ do not correspond to any continuous Painlev\'e equation, and the differential $2 \times 2$ matrix Lax forms of these $d$-Painlev\'e equations have never been known. However, some $6 \times 6$, $4 \times 4$ and $3 \times 3$ matrix Lax forms have been constructed as certain Fuchsian system of differential equations in \cite{Boalch09} for types $d$-$E_8^{(1)}$, $d$-$E_7^{(1)}$ and $d$-$E_6^{(1)}$ respectively. 

On the other hand, It is also known that all the $d$-Painlev\'e equations are obtained from compatibility condition of a linear additive (difference) equation and its additive deformation (e.g. additive matrix type \cite{AB06}, additive scalar type \cite{KNY17,Nagao16}). For example, some 2 $\times$ 2 matrix Lax forms have been obtained utilizing moduli spaces of difference connection on $\mathbb{P}^1$ in \cite{AB06} for type $d$-$E_6^{(1)}$ and $d$-$D_4^{(1)}$.  We call the linear additive (difference) equation the ``{\it additive (difference) Lax form}".  Incidentally, all the scalar differential/discrete (i.e. elliptic-, $q$-, additive) Lax forms for the continuous/discrete Painlev\'e equations have been completely  summarized in \cite{KNY17}.

\subsection{The background of hypergeometric special solutions for $d$-Painlev\'e equations}\label{subsec:sol}　\\

 It is well known that the continuous Painlev\'e equations, namely $P_{\rm II}$, \ldots, $P_{\rm VI}$, admit special solutions expressible in terms of the various hypergeometric functions. The coalescence cascade of hypergeometric functions, from the Gauss hypergeometric function ${}_2F_1$ to the Airy function, corresponds to that of continuous Painlev\'e equations, from $P_{\rm VI}$ to $P_{\rm II}$ in such as \cite{Fuchs1907, IKSY91, KNY17}. Therefore the $d$-Painlev\'e equation of type $D_4^{(1)}$ and its degenerations admit special solutions expressed in terms of the same hypergeometric functions as the continuous Painlev\'e equations. Also the special solutions to the $d$-Painlev\'e equations of type $E_8^{(1)}$, $E_7^{(1)}$ and $E_6^{(1)}$ have been given in terms of the generalized hypergeometric functions $
{}_kF_l$ respectively in \cite{Kajiwara08, KNY17}. As far as we know, determinant formulae of hypergeometric solutions, from type $d$-$E_8^{(1)}$ to $d$-$A_3^{(1)}$, have never been constructed. Incidentally, all the hypergeometric special solutions for the continuous/discrete Painlev\'e equations have been completely summarized in \cite{KNY17}.

 Let us define the additive shifted factorials and the HGF (the generalized hypergeometric series \cite{Bailey35, GaR04}) as follows:
 \begin{equation}\label{eq:Poch}
\begin{array}{l}
(a_1, a_2, \dots, a_i)_j=(a_1)_j(a_2)_j \cdots (a_i)_j ,\quad (a_i)_j=
\displaystyle\prod_{k=0}^{j-1}(a_i+k),
\end{array}
\end{equation}
\begin{equation}\label{eq:HGF}
\begin{array}{l}
{}_kF_l\left(
\begin{array}{ccc}
a_1,&\dots,&a_k\\[0mm]
b_1,&\dots,&b_l
\end{array}
; x
\right)
=\displaystyle\sum_{s=0}^{\infty}\dfrac{(a_1,\ldots,a_k)_s}{(b_1,\ldots,b_l)_s}\dfrac{x^s}{s!}.
\end{array}
\end{equation}
 
Here we consider the condition $k=l+1$. If the parameters $a_k, b_l$ satisfy the relations $a_1+\ldots +a_{l+1}+n=b_1+\ldots+b_l$, the series are called ``{\it $n$-balanced}". If the parameters satisfy the relations $1+a_1=b_1+a_2=\ldots=b_l+a_{l+1}$, $a_2=1+\sfrac{a_1}{2}$, the series are called ``{\it very-well-poised}". 

Then, all the hypergeometric solutions to the $d$-Painlev\'e equations are summarized as in the following degeneration diagram:
 
 \hspace{5mm}
{\arraycolsep=0.1pt
%\[
\begin{equation}\label{eq:solad}\nonumber
\begin{array}{ccccccccccccccccccc}
%{\rm ell.\mbox{($e$-)}}\quad&E^{(1)}_8\\[-2mm]
%&&&&&&&&&&&&&&&{\mathbb Z}\\[-2mm]
%&\downarrow&&&&&&&&&&&&&\nearrow&&\searrow\\
%{\rm mul.}\mbox{($q$-)}\quad&E^{(1)}_8&\rightarrow&E^{(1)}_7&\rightarrow&E^{(1)}_6
%&\rightarrow &D_5^{(1)}&\rightarrow&A_4^{(1)}&\rightarrow&(A_2+A_1)^{(1)}&\rightarrow&(A_1+A_1^{\prime})^{(1)}&\rightarrow&A_1^{(1)}&&{\mathcal D}_6\\[2mm]
%&\downarrow&&\downarrow&&\downarrow&&&\searrow&&\searrow&&\searrow&&\searrow&&\searrow&\\
{\rm add.}\mbox{($d$-)}\quad
&E^{(1)}_8&\rightarrow&E^{(1)}_7&\rightarrow&E^{(1)}_6
&&\rightarrow&&\underset{(\mbox{$P_{\rm VI}$})}{D_4^{(1)}}&\rightarrow&\underset{(\mbox{$P_{\rm V}$})}{A_3^{(1)}}&
\rightarrow
&\underset{(\mbox{$P_{\rm III}$})}{(A_1+A_1^{\prime})^{(1)}}&\rightarrow&A^{(1)}_1&\rightarrow&{{\mathbb Z}_2}\\
&&&&&&&&&&&&\searrow&&\searrow&&&\downarrow\\
&&&&&&&&&&&&&\underset{(\mbox{$P_{\rm IV}$})}{A_2^{(1)}}&\rightarrow&\underset{(\mbox{$P_{\rm II}$})}{A_1^{(1)}}\rightarrow&&\underset{(\mbox{$P_{\rm I}$})}{1}\\
\end{array}
%\]
\end{equation}
}
 
{\arraycolsep=0.1pt
%\[
\begin{equation}\label{eq:classification}\nonumber
\begin{array}{ccccccccccccccccccc}
%{\rm ell.\mbox{($e$-)}}\quad&E^{(1)}_8\\[-2mm]
%&&&&&&&&&&&&&&&{\mathbb Z}\\[-2mm]
%&\downarrow&&&&&&&&&&&&&\nearrow&&\searrow\\
%{\rm mul.}\mbox{($q$-)}\quad&E^{(1)}_8&\rightarrow&E^{(1)}_7&\rightarrow&E^{(1)}_6
%&\rightarrow &D_5^{(1)}&\rightarrow&A_4^{(1)}&\rightarrow&(A_2+A_1)^{(1)}&\rightarrow&(A_1+A_1^{\prime})^{(1)}&\rightarrow&A_1^{(1)}&&{\mathcal D}_6\\[2mm]
%&\downarrow&&\downarrow&&\downarrow&&&\searrow&&\searrow&&\searrow&&\searrow&&\searrow&\\
{\rm sol.}\quad
&\substack{{\rm V.W.P}\\{2- \rm balanced}\\{}_9F_8} \hspace{2mm}&\rightarrow&\substack{{\rm V.W.P}\\{}_7F_6} \hspace{2mm}&\rightarrow&{}_3F_2 \hspace{2mm}
&&\rightarrow&&\underset{{\rm Gauss}}{{}_2F_1}\hspace{2mm}&\rightarrow&\underset{{\rm Kummer}}{{}_1F_1}\hspace{2mm}&
\rightarrow
&{\rm Bessel}\hspace{2mm}&\rightarrow&\times&\rightarrow&\times\\
&&&&&&&&&&&&\searrow&&\searrow&&&\downarrow\\
&&&&&&&&&&&&&\substack{{\rm Hermite}\\{\rm -Weber}}
%\substack{{\rm parabolic}\\{\rm cylinder}}
&\rightarrow&{\rm Airy}&\rightarrow&\times\\
\end{array}
%\]
\end{equation}
}

Here the symbol ${\rm V.W.P}$ means``{\it very-well-poised}". We note that a terminating very-well-poised ${}_7F_6$ can be rewritten as a terminating ${}_4F_3$ (e.g. \cite{Bailey35}) and Kummer (confluent hypergeometric) function ${}_1F_1$ can be also rewritten as ${}_2F_0$ (e.g. \cite{KS98}). 

%
%\noindent
%$\bullet$ Case $d$-$E_7^{(1)}$\\
%
%\noindent
%$\bullet$ Case $d$-$E_6^{(1)}$\\
%
%\noindent
%$\bullet$ Case $d$-$D_4^{(1)}$\\
%
%\noindent
%$\bullet$ Case $d$-$A_3^{(1)}$


% ${}_2F_1$ and Kummer's (confluent hypergeometric) function ${}_1F_1$ (or ${}_2F_0$) in such as \cite{Fuchs1907} and \cite{IKSY91}. It is summarized in \cite{GORS98} that the $d$-Painlev\'e equations of type $D_4^{(1)}$ and $A_3^{(1)}$ have been derived from the compatibility of additive deformation problems (i.e. Schlesinger transformations) for linear equations for continuous Painlev\'e equations of type $P_{\rm VI}$ and $P_{\rm V}$. Due to that, the hypergeometric solutions of type $D_4^{(1)}$ and $A_3^{(1)}$ are naturally expressed in terms of the hypergeometric functions ${}_2F_1$. Also the hypergeometric solutions to the $d$-Painlev\'e equations of type $E_7^{(1)}$ and $E_6^{(1)}$ have given in terms of the generalized hypergeometric functions ${}_4F_3$ and ${}_3F_2$ respectively in \cite{Kajiwara08}.
 
%\subsection{Main results: Hypergeometric special solutions for $d$-Painlev\'e equations}\label{subsec:sol}　\\

\subsection{The background of the Pad\'e method}\label{subsec:pade method}　\\

\noindent
There is a certain connection among Pad\'e approximation/interpolation and Painlev\'e/Garnier systems. The ``{\it Pad\'e method}" gives time evolution equations, scalar Lax pairs and determinant formulae of special solutions simultaneously, by starting from suitable problems of Pad\'e approximation (of differential grid)/interpolation (of difference grid). In \cite{Yamada09-1} the Pad\'e method has been applied to continuous Painlev\'e equations of type $P_{\rm VI}$, $P_{\rm V}$, $P_{\rm IV}$ and the Garnier system\footnote{For the works related to the Garnier system, see \cite{Mano12} (Pad\'e approximation), \cite{MT14,MT17} (Hermite-Pad\'e approximation).
} using differential grid (i.e. Pad\'e approximation) by Y.Yamada.
 
After this, the Pad\'e method for discrete Painlev\'e/Garnier systems\footnote{As the discrete Garnier systems, see \cite{DST13,DT14,OR16-1} (additive Garnier system), \cite{Sakai05, OR16-1} ($q$-Garnier system), \cite{OR16-2} (elliptic Garnier system).} has been applied to the following types: 
\begin{equation}\label{eq:pade-results}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
grid &$q$-Garnier &$e$-$E_8^{(1)}$&$q$-$E_8^{(1)}$&$q$-$E_7^{(1)}$&$q$-$E_6^{(1)}$&
%\multicolumn{2}{|c|}{$q$-$D_5^{(1)}$}
$q$-$D_5^{(1)}$&$q$-$A_4^{(1)}$&$q$-$(A_2+A_1)^{(1)}$\\
\hline\hline
elliptic &&\cite{NTY13}&&&&&&\\
\hline
$q$-quadric & &&\cite{Yamada14}&&&&&\\
\hline
$q$&\cite{NY16, NY17}&&&\cite{Nagao15,Nagao17-2}&\cite{Ikawa13, Nagao15}&\cite{Nagao15}&\cite{Nagao15}&\cite{Nagao15}\\
\hline
differential &\cite{NY16}&&&&\cite{Nagao17-1}&\cite{Ikawa13, Nagao17-1}&\cite{Nagao17-1}&\cite{Nagao17-1}\\
\hline
\end{tabular}
\end{equation} 
Giving a supplementary explanation, in \cite{NY16, NY17} $q$-Garnier system has recently been studied by the Pad\'e method on both differential grid and $q$-grid, and it has become clear that 
 some special solutions of the $q$-Garnier system can be expressed as determinant formulae by using both the $q$-Appell Lauricella function and the generalized $q$-hypergeometric function. 

%It is natural that the continuous/discrete Painlev\'e equations were derived by the interpolation of differential/difference grid respectively.
%only the type $q$-$D_5^{(1)}$ was studied through Pad\'e problems of two kinds of grid, namely $q$-grid in \cite{Nagao15} and differential grid (i.e. Pad\'e approximation) in \cite{Ikawa13}, and 
%Here it may be interesting to note that the Pad\'e approximation method of differential grid (i.e. Pad\'e approximation) has been applied to the type $q$-$D_5^{(1)}$ in \cite{Ikawa13}. In this paper differential grid is applied to type $q$-$E_6^{(1)}$, $q$-$D_5^{(1)}$, $q$-$A_4^{(1)}$ and $q$-$(A_2+A_1)^{(1)}$.

\begin{rem} \label{rem:key}
{\rm {\bf On the key points of the Pad\'e method}

We have two key points on the application for the Pad\'e approximation/interpolation method \cite{Ikawa13, Nagao15, Nagao17-1, Nagao17-2,NY16, NY17, NTY13, Yamada14}. The first key point is how to choose approximated/interpolated functions (e.g. Table (\ref{eq:Ylistad}) and Remark \ref{rem:choice}). The second key point is to consider two linear continuous/difference three term relations (e.g. (\ref{eq:L2L3matrixad}), called ``{\it contiguity relations}") satisfied by the error terms of the Pad\'e approximation/interpolation problems. Then the error terms can be expressed in terms of special solutions of continuous/discrete Painlev\'e/Garnier systems. Therefore the continuous/difference relations (\ref{eq:L2L3matrixad}) are the main subject in the study of the Pad\'e method, and they naturally give the evolution equations, the scalar Lax pairs and the special solutions for the corresponding continuous/discrete Painlev\'e/Garnier systems. \sq
}
\end{rem}

\begin{rem} \label{rem:SOP}
{\rm {\bf On a connection among the Pad\'e method and the theory of semiclassical orthogonal polynomials}

The connection among semiclassical orthogonal polynomials (classical orthogonal polynomials related to a suitable weight function) and Painlev\'e/Garnier systems has been demonstrated in \cite{Magnus95}. It has been shown that coefficients of three term recurrence relations, satisfied by several semiclassical orthogonal polynomials, can be expressed in terms of solutions of Painlev\'e/Garnier systems (see \cite{Clarkson13, Nakazono13, OWF11,Van07,Witte09,Witte15,WO12} for example). Thus there exists a close connection among the Pad\'e method and the theory of semiclassical orthogonal polynomials. Namely, using both approaches, we can obtain the evolution equations, the Lax pairs and the special solutions for the corresponding Painlev\'e/Garnier systems. (The theory of semiclassical orthogonal polynomials is more general and the Pad\'e method is simpler. For example, their relation was briefly proved in \cite{Yamada09-1}.)
\sq
}
\end{rem}

\subsection{The purpose and the organization of this paper}\label{subsec:purpose}　\\

\noindent
The purpose of this paper is to prove Propositions \ref{prop:evolution}, \ref{prop:Lax} and Theorem \ref{thm:sol}, which are the main results derived by applying the Pad\'e interpolation method to type $d$-$E_7^{(1)}$, $d$-$E_6^{(1)}$, $d$-$D_4^{(1)}$ and $d$-$A_3^{(1)}$. 

This paper is organized as follows: In Section \ref{sec:interpolation method} we explain the Pad\'e interpolation method applied to the $d$-Painlev\'e equations, namely the methods for the items (a)--(e) below. In Section \ref{sec:detail results} we present detailed results for type $d$-$E_7^{(1)}$, $d$-$E_6^{(1)}$, $d$-$D_4^{(1)}$ and $d$-$A_3^{(1)}$. In Section \ref{sec:conc} we give a summary and discuss some future problems.

As detailed results given in Section \ref{sec:detail results}, the following items are presented for each type.

\vspace{3mm}
{\bf(a)} Setting of the Pad\'e interpolation problem on additive grid.

{\bf(b)} Contiguity three term relations.

{\bf(c)} The time evolution equation of the $d$-Painlev\'e equation.

{\bf(d)} The additive difference Lax form of scalar type.

{\bf(e)} Determinant formulae of hypergeometric special solutions.\\

%%%%%%%%%%%%%%%
\section{Pad\'e interpolation method on the additive grid}\label{sec:interpolation method}　

\noindent
In this section we explain the methods for deriving the items (a)--(e) in the detailed results given in Section \ref{sec:detail results}. %These contents of the items (b)--(d) below (i.e. Subsection \ref{subsec:itemb}, \ref{subsec:itemc} and \ref{subsec:itemd}) are almost the same as the items (b)--(d) in Section 2 of \cite{Nagao15}.

%%%%%%%%%%%%%%
\subsection{(a) Setting of the Pad\'e interpolation problem on additive grid}\label{subsec:itema}　

\subsubsection{Pad\'e interpolation problem on the additive grid} Let us consider the following interpolation problem on the additive grid:

For a given function $Y(x)$, we look for functions $P_m(x)$ and $Q_n(x)$ which are polynomials of degree $m$ and $n$ $\in \Z_{\geq 0}$, satisfying the interpolation condition 
\begin{equation}\label{eq:padead}
Y(s)=\sfrac{P_m (s)}{Q_n (s)} \quad (s=0,1,\ldots, m+n).
\end{equation}
We call this problem the ``{\it Pad\'e interpolation problem on additive difference grid}", since the interpolation grid $s$ is the additive sequence. Then we call the function $Y(x)$ the ``{\it interpolated function}" in the item (e) below, i.e. Subsection \ref{subsec:iteme}. Correspondingly we call both the polynomials $P_m(x)$ and $Q_n(x)$ ``{\it interpolating polynomials}". The explicit expressions of the polynomials $P_m(x)$ and $Q_n(x)$, which are used in the computations for the item (e) above, are given in the formulae (\ref{eq:Jacobiad}) (see the item (e) below).

\begin{rem}\label{rem:normalizationad}
{\rm {\bf On the common normalization factor of the polynomials $P_m(x)$ and $Q_n(x)$}

The interpolation condition (\ref{eq:padead}) cannot determine the common normalization factor of the interpolating polynomials $P_m(x)$ and $Q_n(x)$. However this normalization factor is not essential for our arguments, i.e. the detailed results in Section \ref{sec:detail results} (see Remark \ref{rem:gaugead}). \sq
}
\end{rem}

\subsubsection{The interpolated function and sequence} 
Let $a_i, b_i, c$ and  $d \in \C^{\times}$ be complex parameters.  
We set up the interpolation problems (\ref{eq:padead}) by specifying the interpolated functions $Y(x)$ and the interpolated sequences $Y_s=Y(s)$ as follows:
\begin{equation}\label{eq:Ylistad}
\begin{tabular}{|c||c|c|c|c|}
\hline
&$d$-$E_7^{(1)}$&$d$-$E_6^{(1)}$&$d$-$D_4^{(1)}$&$d$-$A_3^{(1)}$\\
\hline\hline
$\begin{array}{c}\\Y(x)\\ \\\end{array}$&$\displaystyle\prod_{i=1}^3\dfrac{\Gamma(a_i)\Gamma(x+b_i)}{\Gamma(x+a_i)\Gamma(b_i)}$&
$\displaystyle\prod_{i=1}^2\dfrac{\Gamma (a_i)\Gamma(x+b_i)}{\Gamma(x+a_i)\Gamma(x+b_i)}$&
$c^x\dfrac{\Gamma(a_1)\Gamma(x+b_1)}{\Gamma(x+a_1)\Gamma(b_1)}$&
$d^x\dfrac{\Gamma(x+b_1)}{\Gamma(b_1)}$\\
%&$\dfrac{a_1a_2a_3q^m}{b_1b_2b_3q^n}=1$&&&\\
\hline
$\begin{array}{c}\\Y_s\\ \\ \end{array}$&$\displaystyle\prod_{i=1}^3\dfrac{(b_i)_s}{(a_i)_s}$&
$\displaystyle\prod_{i=1}^2\dfrac{(b_i)_s}{(a_i)_s}$&
$c^s\dfrac{(b_1)_s}{(a_1)_s}$&
$d^s(b_1)_s$\\
%\hline
%parameter&5&4&3&2&1\\
\hline
HGF&${}_4F_3$&${}_3F_2$&${}_2F_1$&${}_2F_0$\\
\hline
\end{tabular}
\end{equation}
where $a_1+a_2+a_3+m-(b_1+b_2+b_3+n)=0$ is a constraint for the parameters in the case $d$-$E_7^{(1)}$. In this paper determinant formulae of special solutions are expressed in terms of the HGF ${}_rF_l$ (hypergeometric functions) given in Table (\ref{eq:Ylistad}), namely the terminating (generalized) hypergeometric functions ${}_4F_3$ (\ref{eq:E7tauad}), ${}_3F_2$ (\ref{eq:E6tauad}), ${}_2F_1$ (\ref{eq:D4tauad}), and ${}_2F_0$ (\ref{eq:A3tauad}). We note that these functions are special solutions to type $d$-$E_7^{(1)}$, $d$-$E_6^{(1)}$, $d$-$D_4^{(1)}$ and $d$-$A_3^{(1)}$ in the second half of Section \ref{subsec:sol}.
%, and the additive shifted factorials are defined by \begin{equation}\label{eq:Poch}
%\begin{array}{l}
%(a_1, a_2, \dots, a_i)_j=(a_1)_j(a_2)_j \cdots (a_i)_j ,\quad (a_i)_j=
%\displaystyle\prod_{k=0}^{j-1}(a_i+k),
%\end{array}
%\end{equation}
%and the HGF (the generalized hypergeometric functions \cite{GaR04}) is defined by
%\begin{equation}\label{eq:HGF}
%\begin{array}{l}
%{}_kF_l\left(
%\begin{array}{ccc}
%a_1,&\dots,&a_k\\[0mm]
%b_1,&\dots,&b_l
%\end{array}
%; x
%\right)
%=\displaystyle\sum_{s=0}^{\infty}\dfrac{(a_1,\ldots,a_k)_s}{(b_1,\ldots,b_l)_s}\dfrac{x^s}{s!}.
%\end{array}
%\end{equation}

%It is well known that the continuous Painlev\'e equations of type $P_{\rm VI}$ and $P_{\rm V}$  admit special solutions expressible in terms of the Gauss's hypergeometric function ${}_2F_1$ and Kummer's (confluent hypergeometric) function ${}_1F_1$ (or ${}_2F_0$) in such as \cite{Fuchs1907} and \cite{IKSY91}. It is summarized in \cite{GORS98} that the $d$-Painlev\'e equations of type $D_4^{(1)}$ and $A_3^{(1)}$ have been derived from the compatibility of additive deformation problems (i.e. Schlesinger transformations) for linear equations for continuous Painlev\'e equations of type $P_{\rm VI}$ and $P_{\rm V}$. Due to that, the hypergeometric solutions of type $D_4^{(1)}$ and $A_3^{(1)}$ are naturally expressed in terms of the hypergeometric functions ${}_2F_1$. Also the hypergeometric solutions to the $d$-Painlev\'e equations of type $E_7^{(1)}$ and $E_6^{(1)}$ have given in terms of the generalized hypergeometric functions ${}_4F_3$ and ${}_3F_2$ respectively in \cite{Kajiwara08}. 
%The functions $Y(x)$ in Table (\ref{eq:Ylistad}) generate the coefficients $p_k$ in power series (\ref{eq:padeYad}) in terms of the terminating (generalized) hypergeometric functions ${}_4F_3$ (\ref{eq:E7pad}) (see Remark \ref{rem:trans}), ${}_3F_2$ (\ref{eq:E6pad}), ${}_2F_1$ (\ref{eq:D4pad}), and ${}_2F_0$ (\ref{eq:A3pad}). 

\begin{rem} \label{rem:choice}
{\rm {\bf On the choice of the interpolated functions $Y(x)$ and sequences $Y_s$}

\noindent
One may wonder how to choose the suitable interpolated functions $Y(x)$ and sequences $Y(x)$. However there is no guiding principle, i.e. only heuristics, to choose the functions $Y(x)$ and the sequences $Y(x)$ in the Pad\'e interpolation method, as far as we know. In this paper we succeed in applying Pad\'e method as follows: we firstly choose the sequences $Y_s$ to type $d$-$E_7^{(1)}$, $d$-$E_6^{(1)}$, $d$-$D_4^{(1)}$ and $d$-$A_3^{(1)}$ by the degeneration limit $q \to 1$ and suitable transformations for the sequences of type $q$-$E_7^{(1)}$, $q$-$E_6^{(1)}$, $q$-$D_5^{(1)}$ and $q$-$A_4^{(1)}$ given in \cite{Nagao15}, and we also determine the functions $Y(x)$ using the well chosen sequences $Y_s$.
%As one of reasons, for example, the functions to type $P_{\rm VI}$, $P_{\rm V}$ and $P_{\rm IV}$ have been already given in \cite{Yamada09-1}. The function to type $D_5^{(1)}$ given in \cite{Ikawa13} is a $q$-analog of the function to type $P_{\rm VI}$ in \cite{Yamada09-1}, however the functions to type $A_4^{(1)}$ and $(A_2+A_1)^{(1)}$ in Table (\ref{eq:Ylistad}) are not $q$-analogs of the functions to type $P_{\rm V}$ and $P_{\rm V}$ in \cite{Yamada09-1}. 
\sq
}
\end{rem}

\subsubsection{Time evolution}
We give the parameter shift operators $T$ as follows:
\begin{equation}\label{eq:Tlistad}
\begin{tabular}{|c||ccc|}
\hline
&parameter&&\\
\hline\hline
$d$-$E_7^{(1)}$&$(a_1,a_2,a_3,b_1,b_2,b_3,m,n)$&$\mapsto$&$(a_1+1,a_2,a_3,b_1+1,b_2,b_3,m,n)$\\
\hline
$d$-$E_6^{(1)}$&$(a_1,a_2,b_1,b_2,m,n)$&$\mapsto$&$(a_1+1,a_2,b_1+1,b_2,m,n)$\\
\hline
$d$-$D_4^{(1)}$&$(a_1,b_1,c,m,n)$&$\mapsto$&$(a_1+1,b_1+1,c,m,n)$\\
\hline
$d$-$A_3^{(1)}$&$(b_1,d,m,n)$&$\mapsto$&$(b_1+1,d,m,n)$\\
\hline
\end{tabular}
\end{equation}
Here the operators $T$ are called the ``{\it time evolutions}", since they specify the directions of the time evolution equations for the corresponding $d$-Painlev\'e equations. 

We consider yet another Pad\'e problem 
%\begin{equation}\label{eq:pade2ad}
$\o{Y}(s)=\sfrac{\o{P}_m (s)}{\o{Q}_n (s)} \quad (s=0,1,\ldots, m+n)$.
%\end{equation}
%where some parameters $a_i, b_i, c, d, m$ and $n$ in the interpolated functions $Y(x)$ and sequences $Y_s$ are shifted. \\
Here for any object $F$ the corresponding shifts are denoted as $\o{F}:=T(F)$ and $\u{F}:=T^{-1}(F)$, and the shift operator $T$ acts on parameters given in Table (\ref{eq:Tlistad}).

%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{(b) Contiguity three term relations}\label{subsec:itemb}　

\subsubsection{Contiguity relations by determinant expressions} 
Let us consider two linear three term relations: $L_2(x)=0$ among $y(x), y(x+1), \o{y}(x)$ and $L_3(x)=0$ among $y(x), \o{y}(x), \o{y}(x-1)$ satisfied by fundamental solutions $y(x)=P_m(x)$, $Y(x)Q_n(x)$, where $L_2$ and $L_3$ are given as expressions 
\begin{align}\label{eq:L2L3matrixad}
L_2(x)\propto
\begin{vmatrix}\nonumber 
y(x) & y(x+1) & \o{y}(x) \\
P_m(x) & P_m(x+1) & \o{P}_m(x)\\
Y(x)Q_n(x) & Y(x+1)Q_n(x+1) & \o{Y}(x)\o{Q}_n(x)
\end{vmatrix}, \\
L_3(x)\propto
\begin{vmatrix} 
y(x) & \o{y}(x) & \o{y}(x-1) \\
P_m(x) & \o{P}_m(x) & \o{P}_m(x-1)\\
Y(x)Q_n(x) & \o{Y}(x)\o{Q}_n(x) & \o{Y}(x-1)\o{Q}_n(x-1)
\end{vmatrix}.
\end{align}
% for unknown function $y(x)$ among $\o{y}(x), y(q x), y(x)$ and $y(x), \o{y}(x), \o{y}(\sfrac{x}{q})$
%, such that their solutions are $y(x)=P_m(x)$, $Y(x)Q_n(x)$, 
Then we call the linear relations $L_2=0$ and $L_3=0$ the ``{\it contiguity relations}", and the contiguity relations are the main subject in our study. \\
%\begin{equation}\label{eq:Casorati}
%\begin{array}{l}
%D_1(x)=
%\begin{vmatrix}\nonumber 
%P_m(x) & P_m(qx) \\
%Y(x)Q_n(x) & Y(qx)Q_n(qx)
%\end{vmatrix}, \\
%D_2(x)=
%\begin{vmatrix} 
%P_m(x) & \o{P}_m(x) \\
%Y(x)Q_n(x) & \o{Y}(x)\o{Q}_n(x)
%\end{vmatrix},\\
%D_3(x)=
%\begin{vmatrix} \nonumber
%\o{P}_m(qx) & \o{P}_m(x/q)\\
%\o{Y}(qx)\o{Q}_n(qx) & \o{Y}(x/q)\o{Q}_n(x/q)\\
%\end{vmatrix}.
%\end{array}
%\end{equation}

%\begin{rem} 
%{\rm {\bf On the monodromy of the solutions $P_m(x)$ and $Y(x)Q_n(x)$}
%
%\noindent
%In the situation of the condition (\ref{eq:padead}) and the functions $Y(x)$ in Table (\ref{eq:Ylistad}), the monodromy of the solutions $P_m(x)$ and $Y(x)Q_n(x)$ in the contiguity relations $L_2=0$ and $L_3=0$ (\ref{eq:L2L3matrixad}) do not depend on parameters shifted by the time evolutions $T$ in Table (\ref{eq:Tlistad}).$\square$
%}
%\end{rem}
\subsubsection{Computation method} 
Let us show the method of computation of the contiguity relations $L_2=0$ and $L_3=0$.\\
We set ${\bf y}(x):=\left[\begin{array}{c}P_m(x)\\Y(x)Q_n(x)\end{array}\right]$ and define Casorati determinants $D_i(x)$ by
\begin{equation}\label{eq:Casoratiad}
\begin{array}{l}
D_1(x):=\det[{\bf y}(x),{\bf y}(x+1)],\hspace{3mm} D_2(x):=\det[{\bf y}(x),{\o{\bf y}}(x)],\hspace{3mm} D_3(x):=\det[{\bf y}(x+1),\o{{\bf y}}(x)].
\end{array}
\end{equation}
Then the expressions (\ref{eq:L2L3matrixad}) can be rewritten as follows:
\begin{equation}\label{eq:L2L3ad}
\begin{array}{l}
L_2(x)\propto D_1(x) \o{y}(x)-D_2(x)y(x+1)+D_3(x)y(x),
\\
L_3(x)\propto \o{D}_1(x-1)y(x)+D_3(x-1) \o{y}(x)-D_2(x)\o{y}(x-1).
\end{array}
\end{equation}
Let us define basic quantities $G(x), K(x)$ and $H(x)$ (e.g. (\ref{eq:E7GKHad}) and (\ref{eq:E6GKHad})) by
\begin{equation}\label{eq:GKHad}
\begin{array}{l}
G(x):=\sfrac{Y(x+1)}{Y(x)},\quad K(x):=\sfrac{\o{Y}(x)}{Y(x)},\quad H(x):={\rm L.C.M}(G_{\rm den}(x), K_{\rm den}(x)).
\end{array}
\end{equation}
Here the abbreviation L.C.M represents the lowest common multiple, and the symbols $G_{\rm den}(x)$ and $G_{\rm num}(x)$ mean the polynomials of the denominator and the numerator in $G(x)$ respectively, and the symbols $K_{\rm den}(x)$ and $K_{\rm num}(x)$ also have the same meaning as $G_{\rm den}(x)$ and $G_{\rm num}(x)$. For example, in the case of $d$-$E_7^{(1)}$, $G_{\rm den}(x)=\prod_{i=1}^3 (x+a_i)$, $G_{\rm num}(x)=\prod_{i=3}^3 (x+b_i)$, $K_{\rm den}(x)=(x+a_1)(x+a_3)$ and $K_{\rm num}(x)=(x+b_1)(x+b_3)$ (see eq. (\ref{eq:E7GKHad})). Substituting these quantities into the determinants (\ref{eq:Casoratiad}), we obtain the following expressions:
\begin{equation}\label{eq:Drelationad}
\begin{array}{l}
D_1(x)=\dfrac{Y(x)}{G_{\rm den}(x)}\left\{G_{\rm num}(x)P_{m}(x)Q_n(x+1)-G_{\rm den}(x)P_{m}(x+1)Q_n(x) \right\},
\\[5mm]
D_2(x)=\dfrac{Y(x)}{K_{\rm den}(x)}\left\{K_{\rm num}(x)P_{m}(x)\o{Q}_n(x)-K_{\rm den}(x)\o{P}_{m}(x)Q_n(x) \right\},
\\[5mm]
D_3(x)=\dfrac{Y(x)}{H(x)}
\left\{\dfrac{H(x)}{K_{\rm den}(x)}K_{\rm num}(x)P_{m}(x+1)\o{Q}_n(x)-\dfrac{H(x)}{G_{\rm den}(x)}G_{\rm num}(x)\o{P}_{m}(x)Q_n(x+1)\right\}.
\end{array}
\end{equation}
Using the interpolation condition (\ref{eq:padead}) and the form of the basic quantities $G(x)$, $K(x)$ and $H(x)$ (e.g. eqs. (\ref{eq:E7GKHad}) and (\ref{eq:E6GKHad})), we can investigate positions of zeros (e.g. $x=0, 1, \ldots, m+n-1$) and degrees of the polynomials (e.g. the polynomial $G_{\rm num}(x)P_{m}(x)Q_n(x+1)-G_{\rm den}(x)P_{m}(x+1)Q_n(x)$ is of degree $m+n+1$ in $x$.) within braces $\{\hspace{3mm}\}$ of the expressions (\ref{eq:Drelationad}). Then we can simply compute the determinants $D_i(x)$ (e.g. eqs. (\ref{eq:E7Dad}) and (\ref{eq:E6Dad}))  except for some factors such as $x-f$, $x-g$ and $c_i$ in $D_i(x)$, where $f$, $g$ and $c_i$ are constants depending on parameters $a_i ,b_i ,c$ but independent of $x$ (see Remark \ref{rem:genericad}). In this way we obtain the contiguity relations $L_2=0$ and $L_3=0$ (e.g. eqs. (\ref{eq:E7L2L3ad}) and (\ref{eq:E6L2L3ad})).

\begin{rem}\label{rem:gaugead}
{\rm {\bf On the gauge invariance of the product $C_0C_1$}

\noindent
Changing the common normalization factor of the interpolating polynomials $P_m(x)$ and $Q_n(x)$, we can induce an $x$-independent gauge transformation of $y(x)$ in the contiguity relations $L_2=0$ and $L_3=0$. Under the $x$-independent gauge transformation of $y(x)$: $y(x)\mapsto Gy(x)$, we can change the coefficients of $\o{y}(x), y(x-1), y(x)$ and $y(x), \o{y}(x), \o{y}(x-1)$ in $L_2=0$ and $L_3=0$ (\ref{eq:L2L3ad}) as follows:
\begin{equation}
\begin{array}{l}
(D_1(x) : D_2(x) : D_3(x))\mapsto(\sfrac{\o{G}D_1(x)}{G} : D_2(x) : D_3(x))\\
(\o{D}_1(x-1) : D_3(x-1) : D_2(x))\mapsto(\sfrac{G\o{D}_1(x-1)}{\o{G}} : D_3(x-1) : D_2(x)).
\end{array}
\end{equation}
Let us define the coefficients $C_0$ and $C_1$ in $L_2=0$ and $L_3=0$ (e.g. eqs. (\ref{eq:E7L2L3ad}) and (\ref{eq:E6L2L3ad})) as the normalization factors of the coefficients of $\o{y}(x)$ and $y(x)$ respectively. Then $C_0$ and $C_1$ change under the gauge transformation, although the product $C_0C_1$ is a gauge invariant quantity. Furthermore, $C_0$ and $C_1$ do not appear in the final form of the $d$-Painlev\'e equations (e.g. eqs. (\ref{eq:E7eqad}) and (\ref{eq:E6eqad})). \sq
}
\end{rem}

\begin{rem}\label{rem:genericad}
{\rm {\bf On two meanings of the variables $f, g$ and parameters $m, n$}

\noindent
We use $f$ and $g$ for two different meanings. The first meaning is constants (i.e. special solutions) $f$ and $g$ which are explicitly determined in terms of parameters $a_i, b_i, m$ and $n$ by the Pad\'e interpolation problem (e.g. eqs. (\ref{eq:E7Dad}), (\ref{eq:E7L2L3ad}), (\ref{eq:E7solad}), (\ref{eq:E6Dad}), (\ref{eq:E6L2L3ad}) and (\ref{eq:E6solad})). The second meaning is generic variables (i.e. generic solutions) $f$ and $g$ $\in \P^1$ apart from the Pad\'e interpolation problem (e.g. eqs. (\ref{eq:E7eqad}), (\ref{eq:E7L1L2ad}), (\ref{eq:E6eqad}) and (\ref{eq:E6L1L2ad})), namely $f$ and $g$ are unknown functions in the $d$-Painlev\'e equation. In the items (c), (d) (resp. in the items (b) and (e)) we consider $f$ and $g$ in the second meaning (resp. in the first meaning).

Similarly we use $m$ and $n$ for two meanings. In the first meaning, $m$ and $n$ $\in \Z_{\geq 0}$ are non-negative integer parameters (e.g. eqs. (\ref{eq:E7Yad}), (\ref{eq:E7Dad}), (\ref{eq:E7L2L3ad}), (\ref{eq:E6Yad}), (\ref{eq:E6Dad}) and (\ref{eq:E6L2L3ad})). In the second meaning, $m$ and $n$ $\in \C^{\times}$ are generic complex parameters, namely $m$ and $n$ are replaced by generic complex parameters $a_0$ and $b_0$ $\in \C^{\times}$ respectively 
(e.g. eqs. (\ref{eq:E7eqad}), (\ref{eq:E7L1L2ad}), (\ref{eq:E6eqad}) and (\ref{eq:E6L1L2ad})). In the items (c) and (d) (resp. in the items (a), (b) and (e)), we consider $f$ and $g$ in the second meaning (resp. in the first meaning). Then the result of the compatibility of the contiguity relations $L_2=0$ and $L_3=0$ also holds with respect to the second meaning. \sq
}
\end{rem}

%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{(c) The time evolution of the $d$-Painlev\'e equation}\label{subsec:itemc}　\\

\noindent
The computation method is as follows: Let us consider generic variables $f, g \in \P^1$ and generic parameter $a_0, b_0 \in \C^{\times}$ as in the second meaning in Remark \ref{rem:genericad}. Then we can derive the $d$-Painlev\'e equation as the necessary condition for the compatibility of the contiguity relations $L_2=0$ and $L_3=0$ (e.g. eqs. (\ref{eq:E7L2L3ad}) and (\ref{eq:E6L2L3ad})). Computing the compatibility condition, we determine three variables $\u{g}, \o{f}$ and $C_0C_1$. Expressions for variables $\u{g}$ and $\o{f}$ are obtained in terms of variables $f$ and $g$. An expression for the product $C_0C_1$ is obtained in terms of variables $f, g$ and $\o{f}$ (and hence in terms of variables $f$ and $g$).

Finally we note as follows: The first and the second expressions are the $d$-Painlev\'e equation (e.g. eqs. (\ref{eq:E7eqad}) and (\ref{eq:E6eqad})). The third expression is a constraint for the product $C_0C_1$ (e.g. eqs. (\ref{eq:E7C0C1ad}) and (\ref{eq:E6C0C1ad})). 

\begin{prop}\label{prop:evolution}
Applying the calculation method above, we can derive the time evolutions for type $d$-$E_7^{(1)}$ (\ref{eq:E7eqad}), $d$-$E_6^{(1)}$ (\ref{eq:E6eqad}), $d$-$D_4^{(1)}$ (\ref{eq:D4eqad}) and $d$-$A_3^{(1)}$ (\ref{eq:A3eqad}), which are given in item {\rm (c)} of Section \ref{sec:detail results}.
\sq
\end{prop}

Furthermore, due to the method, we also construct scalar Lax pairs and determinant formulae of hypergeometric special solutions, while simultaneously deriving these time evolutions, as in Section \ref{subsec:itemd} and \ref{subsec:iteme}.

%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{(d) The additive difference Lax form of scalar type}\label{subsec:itemd}　

\subsubsection{Scalar Lax pair}
Let us consider a linear three term equations for the unknown function $y(x)$: $L_1(x)=0$ among $y(x+1), y(x), y(x-1)$ and its deformation equation $L_2(x)=0$ among $y(x), y(x+1), \o{y}(x)$, where $L_1$ and $L_2$ are given as expressions 
\begin{equation}\label{eq:L1L2ad}
\begin{array}{l}
L_1(x)=A_1(x)y(x-1)+A_2(x)y(x)+A_3(x)y(x+1),
\\
L_2(x)=A_4(x)\o{y}(x)+A_5(x)y(x)+A_6(x)y(x-1).
\end{array}
\end{equation}  
We call the linear three term equation $L_1=0$ and its deformation equation $L_2=0$ (\ref{eq:L1L2ad}) the ``{\it scalar Lax pair}", if the compatibility condition of the linear equations $L_1=0$ and $L_2=0$ (\ref{eq:L1L2ad}) is equivalent to a $d$-Painlev\'e equation. We note that the scalar Lax pair $L_1=0$ and $L_2=0$ is equivalent to the pair of contiguity relations $L_2=0$ and $L_3=0$.\\

\subsubsection{Computation method}
Let us show how to compute the scalar Lax pair. Similarly to the item (c), we consider generic variables $f, g \in \P^1$ and generic parameters $a_0, b_0 \in \C^{\times}$ as in the second meaning in Remark \ref{rem:genericad}. We derive the Lax pair $L_1=0$ and $L_2=0$, which satisfies the compatibility condition, by using the results of the items (a)--(c) as follows: Firstly the Lax equation $L_2=0$ (e.g. eqs. (\ref{eq:E7L1L2ad}) and (\ref{eq:E6L1L2ad})) in the item (d) is the same as the contiguity relation $L_2=0$ (e.g. eqs. (\ref{eq:E7L2L3ad}) and (\ref{eq:E6L2L3ad})) in the item (b) under an $x$-independent gauge transform of $y(x)$ and changes of parameters. Secondly the Lax equation $L_1=0$ can be obtained as follows: Combining the contiguity relations $L_2=0$ and $L_3=0$ (e.g. eqs. (\ref{eq:E7L2L3ad}) and (\ref{eq:E6L2L3ad})) under generic variables $f, g$ and generic parameters $a_0, b_0$, one obtains a linear equation among the three terms $y(x+1), y(x)$ and $y(x-1)$ (see the figure below), whose coefficient functions depend on the variables $f, g, \o{f}, C_0$ and $C_1$. However the variables $C_0$ and $C_1$ appear through the product $C_0C_1$. Therefore expressing $\o{f}$ (e.g. eqs. (\ref{eq:E7eqad}), (\ref{eq:E6eqad})) and $C_0C_1$ (e.g. eqs. (\ref{eq:E7C0C1ad}) and (\ref{eq:E6C0C1ad})) in terms of $f$ and $g$ only, one obtains the Lax equation $L_1=0$ (e.g. eqs. (\ref{eq:E7L1L2ad}) and (\ref{eq:E6L1L2ad})). 
%\begin{figure}[http] 
\begin{center}\setlength{\unitlength}{1.2mm}
\begin{equation}\label{eq:compatibility}
\begin{picture}(50,28)(-5,-3)
\put(-1,23){$\o{y}(x-1)$}
\put(20,23){$\o{y}(x)$}
\put(41,23){$\o{y}(x+1)$}
\put(-1,-4){$y(x-1)$}
\put(20,-4){$y(x)$}
\put(41,-4){$y(x+1)$}
\put(1,4){$L_2(x-1)$}
\put(26,4){$L_2(x)$}
\put(11,14){$L_3(x)$}
\put(29,14){$L_3(x+1)$}
%\put(11,-4){$L_1(x)$}
%\put(0,-1){\line(1,0){43}}
\put(0,0){\line(1,0){20}}
\put(0,0){\line(0,1){20}}
\put(0,20){\line(1,-1){20}}
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\put(21,1){\line(0,1){20}}
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\put(23,21){\line(1,0){20}}
\put(43,21){\line(0,-1){20}}
%\put(0,22){\line(1,0){43}}
\end{picture}
\end{equation}
\end{center}

Then, under generic variables $f, g$ and generic parameters $a_0, b_0$, the $d$-Painlev\'e equation (e.g. eqs. (\ref{eq:E7eqad}) and (\ref{eq:E6eqad})) is sufficient for the compatibility of the Lax pair $L_1=0$ and $L_2=0$ (e.g. eqs. (\ref{eq:E7L1L2ad}) and (\ref{eq:E6L1L2ad})). The proof for case of $d$-$E_7^{(1)}$ will be shown in Appendix \ref{sec:Proof}. Other cases of $d$-$E_6^{(1)}$, $d$-$D_4^{(1)}$ and $d$-$A_4^{(1)}$ are similarly proved. 

\begin{prop}\label{prop:Lax}
Computing by the method mentioned above, we can derive the additive difference Lax pairs of scalar form for type $d$-$E_7^{(1)}$ (\ref{eq:E7L1L2ad}), $d$-$E_6^{(1)}$ (\ref{eq:E6L1L2ad}), $d$-$D_4^{(1)}$ (\ref{eq:D4L1L2ad}) and $d$-$A_3^{(1)}$ (\ref{eq:A3L1L2ad}), which are obtained in item {\rm (d)} of Section \ref{sec:detail results}. $\square$
\end{prop}

%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{(e) Determinant formulae of hypergeometric special solutions}\label{subsec:iteme}　\\

\noindent
By construction, expressions for $f$ and $g$ as in the first meaning in Remark \ref{rem:genericad} give a special solution for the $d$-Painlev\'e equation.
We present how to compute determinant formulae of the special solutions.

\subsubsection{Cauchy-Jacobi formulae on additive grid} We derive formulae (\ref{eq:Jacobiad}) which are convenient for computing the special solutions $f$ and $g$. The {\it Cauchy--Jacobi formulae} (\ref{eq:Jacobi}) are essentially presented in \cite{Jacobi}. For a given sequence $Y_s$, the polynomials $P_m(x)$ and $Q_n(x)$ of degree $m$ and $n$ for an interpolation problem 
\begin{equation} \label{eq:pade}
Y_s=\sfrac{P_m (x_s)}{Q_n (x_s)} \quad (s=0, 1, \dots, m+n)
\end{equation}
are given by the following determinant expressions:
\begin{equation}\label{eq:Jacobi}
P_{m}(x)=F(x)\det\Big[\sum^{m+n} _{s=0}u_s \dfrac{x_s^{i+j}}{x-x_s }\Big]^n _{i,j=0}, \quad Q_n(x)=\det\Big[\sum^{m+n} _{s=0}u_s x_s^{i+j}(x-x_s )\Big]^{n-1} _{i,j=0},
\end{equation}
where $u_s =\sfrac{Y_s}{F^{\prime}(x_s)}$ and $F(x)=\prod_{i=0}^{m+n}(x-x_i )$.

In the additive grid case of problem (\ref{eq:pade}) (i.e., the case of problem (\ref{eq:padead})), the formulae  (\ref{eq:Jacobi}) take the following form:
\begin{equation}\label{eq:Jacobiad}
\begin{array}{l}
P_{m}(x)=\dfrac{F(x)}{(-(m+n))_{m+n}^{n+1}}\det\Big[\displaystyle\sum^{m+n} _{s=0}Y_s\dfrac{(-(m+n))_s }{s!}\dfrac{s^{i+j}}{x-s}\Big]^n _{i,j=0},
\\
Q_n(x)=\dfrac{1}{(-(m+n))_{m+n}^{n}}\det\Big[\displaystyle\sum^{m+n} _{s=0}Y_s\dfrac{(-(m+n))_s }{s!}s^{i+j}(x-s)\Big]^{n-1} _{i,j=0}.
\end{array}
\end{equation}

In the derivation of (\ref{eq:Jacobiad}), we have used the following relations:
\begin{align}\label{eq:Fprime}
F^{\prime}(x_s)=&(x_s -x_0 )\dots(x_s -x_{s-1})(x_s -x_{s+1})\dots(x_s -x_{m+n})\nonumber \\
=&(s-0)(s-1)\cdots \{s-(s-1)\}\{s-(s+1)\}\cdots \{s-(m+n)\}\nonumber\\
=&\sfrac{s!(-(m+n))_{m+n}}{(-(m+n))_s}.  
\end{align}
Moreover, substituting the values of $Y_s$ (\ref{eq:Ylistad}) and $F^{\prime}(x_s )$ (\ref{eq:Fprime}) into the formulae (\ref{eq:Jacobi}), one obtains the determinant formulae (\ref{eq:Jacobiad}).

\subsubsection{Computation method}
Let us demonstrate how to compute the special solutions $f$ and $g$. We can derive the expressions for the special solutions $f$ and $g$ by comparing the determinants $D_i(x)$ in eq. (\ref{eq:Drelationad}) and $D_i(x)$ (e.g. eqs. (\ref{eq:E7Dad}) and (\ref{eq:E6Dad})) in the item (b) as the identity with respect to the variable $x$ and by applying the formulae (\ref{eq:Jacobiad}). 

In case of $d$-$E_7^{(1)}$,  we make the following calculation: Firstly substituting $x=-a_i$ $(i=1, 2)$ into the determinants $D_1(x)$ in eq. (\ref{eq:Drelationad}) and $D_1(x)$ in (\ref{eq:E7Dad}) respectively, we construct an expression for the special solution $f$ in the first equation of eq. (\ref{eq:E7solad}) by comparing the two expressions for $D_1(x)$ and applying the formulae (\ref{eq:Jacobiad}). Similarly substituting $x=-b_i$ $(i=2, 3)$ into the determinants $D_3(x)$ in eq. (\ref{eq:Drelationad}) and $D_3(x)$ in eq. (\ref{eq:E7Dad}) respectively, we construct an expression for the special solution $g$ in the second equation of eq. (\ref{eq:E7solad}) by comparing the two expressions for $D_3(x)$ and by applying the formulae (\ref{eq:Jacobiad}).

\begin{thm}\label{thm:sol}
By the computation method above, we can construct determinant formulae of hypergeometric special solutions as the explicit forms for type $d$-$E_7^{(1)}$ (\ref{eq:E7solad}), $d$-$E_6^{(1)}$ (\ref{eq:E6solad}), $d$-$D_4^{(1)}$ (\ref{eq:D4solad}) and $d$-$A_3^{(1)}$ (\ref{eq:A3solad}), which are derived in item {\rm (e)} of Section \ref{sec:detail results}. 
\sq
\end{thm}

%\begin{rem}{\bf On degenerations from hypergeometric special solutions for $q$-Painlev\'e equations}
%\sq
%\end{rem}

%%%%%%%%%%%%%%%%%%%%%%%%
\section{Detailed results}\label{sec:detail results}　\\

\noindent
In this section, for each case $d$-$E_7^{(1)}$,  $d$-$E_6^{(1)}$, $d$-$D_4^{(1)}$ and $d$-$A_3^{(1)}$, we show the detailed results derived by the method which was explained in Section \ref{sec:interpolation method}.

We use the following notations:
\begin{equation}\label{eq:notationad}
\begin{array}{l}
\sfrac{a_1a_2\dots a_n}{b_1b_2\dots b_n}:=\dfrac{a_1a_2\dots a_n}{b_1b_2\dots b_n},\quad {\mathcal N}(x):=\prod_{i=0}^{m+n-1}(x-i),\\[3mm]
T_{a_i}(F):=F |_{a_i \to a_i+1},\quad T_{a_i}^{-1}(F):= F |_{a_i \to a_i-1},
%\\ \o{F}=T(F),\quad \u{F}=T^{-1}(F),
\end{array}
\end{equation}
for any quantity (or function) $F$ depending on variables $a_i$ and $b_i$.

%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Case $d$-$E_7^{(1)}$}\label{subsec:E7ad}　\\

\noindent
{\bf(a)} Setting of the Pad\'e interpolation problem on the additive grid\\

%E7ad
In Table (\ref{eq:Ylistad}) the interpolated function, the interpolated sequence and the constraint for the parameters are set up as
\begin{equation}\label{eq:E7Yad}
Y(x):=\displaystyle\prod_{i=1}^3\dfrac{\Gamma(a_i)\Gamma(x+b_i)}{\Gamma(x+a_i)\Gamma(b_i)}, \quad Y_s=\displaystyle\prod_{i=1}^3\dfrac{(b_i)_s}{(a_i)_s}, \quad m+\sum_{i=1}^3 a_i =n+\sum_{i=1}^3 b_i, 
\end{equation}
and in Table (\ref{eq:Tlistad}) the time evolution is chosen as
\begin{equation}\label{eq:E7Tad}
T: (a_1,a_2,a_3,b_1,b_2,b_3,m,n) \mapsto (a_1+1,a_2,a_3+1,b_1+1,b_2,b_3+1,m,n).
\end{equation}
\\

%E7ad
\noindent
{\bf(b)} Contiguity three term relations\\

By Definition (\ref{eq:GKHad}) we have the basic quantities
\begin{equation}\label{eq:E7GKHad}
\displaystyle G(x)=\prod_{i=1}^{3}\dfrac{(x+b_i)}{(x+a_i)},\quad 
K(x)=\prod_{i=1, 3}\dfrac{a_i(x+b_i)}{b_i(x+a_i)},\quad H(x)=b_1b_3\prod_{i=1}^{3}(x+a_i),
\end{equation}
and by the expression (\ref{eq:Drelationad}) we obtain the Casorati determinants
\begin{equation}\label{eq:E7Dad}
\begin{array}{l}
D_1(x)=\sfrac{c_0(x-f){\mathcal N}(x)Y(x)}{G(x)_{\rm den}},\quad
D_2(x)=\sfrac{c_1(x-h)(x-m-n){\mathcal N}(x)Y(x)}{K(x)_{\rm den}},
\\[5mm]
D_3(x)=\sfrac{c_1(x-g){\mathcal N}(x)Y(x)\prod_{i=1,3}(x+b_i)}{H(x)},
\end{array}
\end{equation}
where $h=g+a_2-b_1-b_3-n$. Here $f$, $g$, $c_0$ and $c_1$ are constants depending on parameters $a_i, b_i \in \C^{\times}(i=1,2,3)$ and $m,n \in \Z_{\geq 0}$ but independent of $x$. Then the contiguity relations $L_2=0$ and $L_3=0$ are expressed by
\begin{equation}\label{eq:E7L2L3ad}
\begin{array}{l}
L_2(x)=C_0(f-x)\o{y}(x)-(x+a_2)(x-m-n)(x-h)y(x+1)
+(x+b_1)(x+b_3)(x-g)y(x),\\[5mm]
L_3(x)=C_1(x-\o{f}-1)y(x)+\displaystyle\prod_{i=1,3}(x+a_i)(x-g-1)\o{y}(x)
-x(x+b_2-1)(x-h)\o{y}(x-1),
\end{array}
\end{equation}
where $C_0=\sfrac{b_1b_3c_0}{c_1}$ and $C_1=\sfrac{a_1a_3\o{c}_0}{c_1}$.

Take note that in the items (c) and (d) below we study the contiguity relations $L_2=0$ and $L_3=0$ (\ref{eq:E7L2L3ad}) for generic complex parameters $a_0, b_0$ (replacing considering $m, n \in \Z_{\geq 0}$ by $a_0, b_0 \in \C^{\times}$) and generic variables $f, g \in \P^1$ (depending on parameters $a_i, b_i \in \C^{\times}, i=0,1,2,3$) apart from the Pad\'e interpolation problem (\ref{eq:padead}) with eqs. (\ref{eq:E7Yad}) and (\ref{eq:E7Tad}) (see Remark \ref{rem:genericad}).\\

%E7ad
\noindent
{\bf(c)} The time evolution of the $d$-Painlev\'e equation\\

Compatibility of the contiguity relations $L_2=0$ and $L_3=0$ (\ref{eq:E7L2L3ad}) gives the evolution equations and the constraint on the product $C_0C_1$ as follows:
\begin{equation}\label{eq:E7eqad}
\begin{array}{l}
\dfrac{(f-h)(f-\u{h}+1)}{(f-g)(f-\u{g})}=\dfrac{A_2(f)}{A_1(f)}, \quad \dfrac{(f-h)(\o{f}-h+1)}{(f-g)(\o{f}-g)}=\dfrac{A_2(h)}{A_1(g)}
\end{array}
\end{equation}
and
\begin{equation}\label{eq:E7C0C1ad}
\begin{array}l
C_0C_1=\sfrac{(h-g)(h-g-1)A_1(g)}{(f-g)(\o{f}-g)}\\
\phantom{C_0C_1}=\sfrac{(h-g)(h-g-1)A_2(h)}{(f-h)(\o{f}-h+1)}
\end{array}
\end{equation}
where $A_1(x)=(x+a_2)(x+b_2)(x+1)(x-a_0-b_0)$ and $A_2(x)=\prod_{i=1,3}(x+a_i)(x+b_i)$.

The evolution equations (\ref{eq:E7eqad}) are equivalent to the $d$-Painlev\'e equation of type $E_7^{(1)}$ given in \cite{Kajiwara08, KNY17, Nagao16, RGTT01, Sakai07}.
%When the coordinates of $f$ and $g$ are changed into $\sfrac{1}{f}$ and $\sfrac{1}{g}$ respectively, 
The 8 singular points in coordinates $(f,g)$ are on the two lines $f=g$ and $f=h(=g+a_2-b_1-b_3-b_0)$ as follows:
\begin{equation}\label{eq:E78pad}
\begin{array}{l}
(f,g)=(-a_2,-a_2), (-b_2,-b_2), (-1,-1), (-a_0-b_0,-a_0-b_0), \\
\phantom{(f,g)=}(-a_1,a_3-b_2+a_0), (-b_3, b_1-a_2+b_0), (-a_3,a_1-b_2+a_0), (-b_1, b_3-a_2+b_0).
\end{array}
\end{equation}

%E7ad
\noindent
{\bf(d)} The additive difference Lax form of scalar type\\

The contiguity relations $L_2=0$ and $L_3=0$ (\ref{eq:E7L2L3ad}) give two scalar additive Lax equations $L_1=0$ and $L_2=0$ expressed by
\begin{equation}\label{eq:E7L1L2ad}
\begin{array}{l}
L_1(x)=(g-h)\Big[\dfrac{A_1(g)}{(f-g)(x-g-1)}-\dfrac{A_2(h)}{(f-h)(x-h))}\Big]y(x)
\\[5mm]\phantom{L_1(x)}
-\dfrac{x\prod_{i=1}^3(x+b_i-1)}{x-f-1}\Big[y(x-1)-\dfrac{(x-h-1)(x-a_0-b_0-1)(x+a_2-1)}{(x-g-1)(x+b_1-1)(x+b_3-1)}y(x)\Big]
\\[5mm]\phantom{L_1(x)}
-\dfrac{(x-a_0-b_0)\prod_{i=1}^3(x+a_i)}{x-f}\Big[y(x+1)-\dfrac{(x-g)(x+b_1)(x+b_3)}{(x-h)(x-a_0-b_0)(x+a_2)}y(x)\Big],
\\[5mm]
L_2(x)=(f-x)\o{y}(x)-(x+a_2)(x-h)(x-a_0-b_0)y(x+1)
+(x+b_1)(x+b_3)(x-g)y(x), 
\end{array}
\end{equation}
where $h$ and $A_i(x)$ are given in the Casorati determinant  (\ref{eq:E7Dad}) and the contiguity relations (\ref{eq:E7L2L3ad}).

The additive Lax form of scalar type $L_1=0$ and $L_2=0$ (\ref{eq:E7L1L2ad}) is equivalent to the scalar ones in \cite{KNY17,  Nagao16} by using suitable gauge transformations of $y(x)$. On the other hand, the differential $4 \times 4$ matrix Lax form has been given as certain Fuchsian system of differential equations in \cite{Boalch09}.\\

\noindent
%E7ad
{\bf(e)} Determinant formulae of hypergeometric special solutions

The hypergeometric solutions are constructed as the following explicit forms: 
\begin{equation}\label{eq:E7solad}
\begin{array}{l}
\dfrac{f+a_i}{f+b_j}=\alpha\dfrac{T_{a_i}(\tau_{m,n})T_{a_i}^{-1}(\tau_{m+1,n-1})}
{T_{b_j}^{-1}(\tau_{m,n})T_{b_j}(\tau_{m+1,n-1})} \quad  (i, j=1,2,3),\quad 
\dfrac{g+a_2}{h+a_1}=\beta\dfrac{T_{a_2}(\o{\tau}_{m,n})T_{a_2}^{-1}(\tau_{m+1,n-1})}
{T_{a_1}(\tau_{m,n})T_{a_1}(\o{\tau}_{m+1,n-1})},
\end{array}
\end{equation}
where the determinant $\tau_{m,n}$ is given by
\begin{equation}\label{eq:E7tauad}
\begin{array}l
\ds \tau_{m,n}=\det\Big[(b_1)_i(a_1-j)_j{}_4F_3 \Big(\substack{\displaystyle{b_1+i, b_2, b_3 ,-(m+n)}\\[3mm]{\displaystyle{a_1-j, a_2, a_3}}},1\Big)\Big]^n _{i,j=0}, \\[5mm]
\alpha=-\dfrac{(a_i+m+n)(a_i-1)^n(b_j-1)^n}{a_i^{n+1}b_j^n}\dfrac{\prod_{k=1}^3(b_k-a_i)}{\prod_{k=1}^3(a_k-b_j)},\\[5mm]
\beta=\dfrac{b_1b_3(a_2+m+n)(a_2-1)^n(b_2-a_2)}{a_2^{n+1}a_3(b_1-a_1)(b_3-a_1)}.
\end{array}
\end{equation}
  
These determinant formulae of the generalized hypergeometric solutions (\ref{eq:E7solad}) are given in terms of the terminating hypergeometric series ${}_4F_3$. A certain $1 \times 1$ determinant formula has been constructed in terms of a non-terminating very-well-poised ${}_7F_6$ in \cite{Kajiwara08,KNY17}. Again, we note that a terminating very-well-poised ${}_7F_6$ and a terminating ${}_4F_3$ are rewritten with each other in \cite{Bailey35}. The general determinant formulae have not been given yet in terms of a non-terminating ${}_4F_3$ (or ${}_7F_6$) as far as we know.

%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Case $d$-$E_6^{(1)}$}\label{subsec:E6ad}　\\

\noindent
{\bf(a)} Setting of the Pad\'e interpolation problem on the additive grid\\

%E6ad
In Table (\ref{eq:Ylistad}) the interpolated function and the interpolated sequence are set up as
\begin{equation}\label{eq:E6Yad}
Y(x):=\displaystyle\prod_{i=1}^2\dfrac{\Gamma(a_i)\Gamma(x+b_i)}{\Gamma(x+a_i)\Gamma(b_i)}, \quad Y_s=\displaystyle\prod_{i=1}^2\dfrac{(b_i)_s}{(a_i)_s},
\end{equation}
and in Table (\ref{eq:Tlistad}) the time evolution is chosen as
\begin{equation}\label{eq:E6Tad}
T: (a_1,a_2,b_1,b_2,m,n) \mapsto (a_1+1,a_2,b_1+1,b_2,m,n).
\end{equation}
\\

%E6ad
\noindent
{\bf(b)} Contiguity three term relations\\

By Definition (\ref{eq:GKHad}) we have the basic quantities
\begin{equation}\label{eq:E6GKHad}
\displaystyle G(x)=\prod_{i=1}^{2}\dfrac{(x+b_i)}{(x+a_i)},\quad 
K(x)=\dfrac{a_1(x+b_1)}{b_1(x+a_1)},\quad H(x)=b_1\prod_{i=1}^{2}(x+a_i),
\end{equation}
and by the expression (\ref{eq:Drelationad}) we obtain the Casorati determinants
\begin{equation}\label{eq:E6Dad}
\begin{array}{l}
D_1(x)=\sfrac{c_0(x-f){\mathcal N}(x)Y(x)}{G(x)_{\rm den}},\quad
D_2(x)=\sfrac{c_1(x-m-n){\mathcal N}(x)Y(x)}{K(x)_{\rm den}},
\\[5mm]
D_3(x)=\sfrac{c_1(x-g){\mathcal N}(x)Y(x)(x+b_1)}{H(x)},
\end{array}
\end{equation}
where $f, g, c_0$ and $c_1$ are constants depending on parameters $a_i, b_i \in \C^{\times}(i=1,2)$ and $m,n \in \Z_{\geq 0}$ but independent of $x$. Then the contiguity relations $L_2=0$ and $L_3=0$ are expressed by
\begin{equation}\label{eq:E6L2L3ad}
\begin{array}{l}
L_2(x)=C_0(f-x)\o{y}(x)-(x-m-n)(x+a_2)y(x+1)+(x+b_1)(x-g)y(x),\\
L_3(x)=C_1(x-\o{f}-1)y(x)+(x+a_1)(x-g-1)\o{y}(x)
-x(x+b_2-1)(x-h)\o{y}(x-1),
\end{array}
\end{equation}
where $C_0=\sfrac{b_1c_0}{c_1}$ and $C_1=\sfrac{a_1\o{c}_0}{c_1}$.\\

Take note that in the items (c) and (d) below we study the contiguity relations $L_2=0$ and $L_3=0$ (\ref{eq:E6L2L3ad}) for generic complex parameters $a_0, b_0$ (replacing considering $m, n \in \Z_{\geq 0}$ by $a_0, b_0 \in \C^{\times}$) and generic variables $f, g \in \P^1$ (depending on parameters $a_i, b_i \in \C^{\times}, i=0,1,2$) apart from the Pad\'e interpolation problem (\ref{eq:padead}) with eqs. (\ref{eq:E6Yad}) and (\ref{eq:E6Tad}) (see Remark \ref{rem:genericad}).\\

%E6ad
\noindent
{\bf(c)} The time evolution of the $d$-Painlev\'e equation\\

Compatibility of the contiguity relations $L_2=0$ and $L_3=0$ (\ref{eq:E6L2L3ad}) gives the evolution equations and the constraint on the product $C_0C_1$ as follows:
\begin{equation}\label{eq:E6eqad}
\begin{array}{l}
(f-g)(f-\u{g})=\dfrac{(f+a_2)(f+b_2)(f+1)(f-a_0-b_0)}{(f+a_1)(f+b_1)},\\
(f-g)(\o{f}-g)=\dfrac{(g+a_2)(g+b_2)(g+1)(g-a_0-b_0)}{(g-a_1+b_2-a_0)(g+a_2-b_1-b_0)}
\end{array}
\end{equation}
and
\begin{equation}\label{eq:E6C0C1ad}
\begin{array}l
C_0C_1=\sfrac{(g+a_2)(g+b_2)(g+1)(g-a_0-b_0)}{(f-g)(\o{f}-g)}\\
\phantom{C_0C_1}=(g-a_1+b_2-a_0)(g+a_2-b_1-b_0).
\end{array}
\end{equation}

The evolution equations (\ref{eq:E6eqad}) are equivalent to the $d$-Painlev\'e equation of type $E_6^{(1)}$ given in \cite{Kajiwara08, KNY17, Nagao16, RGTT01, Sakai07}.
%When the coordinates of $f$ and $g$ are changed into $\sfrac{1}{f}$ and $\sfrac{1}{g}$ respectively, 
The 8 singular points in coordinates $(f,g)$ are on the three lines $f=g$, $f=\infty$ and $g=\infty$ as follows:
\begin{equation}\label{eq:E68pad}
\begin{array}{l}
(f,g)=(-a_2,-a_2), (-b_2,-b_2), (-1,-1), (a_0+b_0, a_0+b_0), \\
\phantom{(f,g)=}(\infty,a_3-b_2+a_0), (\infty, b_1-a_2+b_0), (-a_1,\infty), (-b_1, \infty).
\end{array}
\end{equation}

%E6ad
\noindent
{\bf(d)} The additive difference Lax form of scalar type\\

The contiguity relations $L_2=0$ and $L_3=0$ (\ref{eq:E6L2L3ad}) give two scalar additive Lax equations $L_1=0$ and $L_2=0$ expressed by
\begin{equation}\label{eq:E6L1L2ad}
\begin{array}{l}
L_1(x)=\Big[\dfrac{(g+a_2)(g+b_2)(g+1)(g-a_0-b_0)}{(f-g)(x-g-1)}-(g-a_1+b_2-a_0)(g+a_2-b_1-b_0)\Big]y(x)
\\[5mm]\phantom{L_1(x)}
-\dfrac{x\prod_{i=1}^2(x+b_i-1)}{x-f-1}\Big[y(x-1)-\dfrac{(x-a_0-b_0-1)(x+a_2-1)}{(x-g-1)(x+b_1-1)}y(x)\Big]
\\[5mm]\phantom{L_1(x)}
-\dfrac{(x-a_0-b_0)\prod_{i=1}^2(x+a_i)}{x-f}\Big[y(x+1)-\dfrac{(x-g)(x+b_1)}{(x-a_0-b_0)(x+a_2)}y(x)\Big],
\\[5mm]
L_2(x)=C_0(f-x)\o{y}(x)-(x-a_0-b_0)(x+a_2)y(x+1)+(x+b_1)(x-g)y(x).
\end{array}
\end{equation}

The additive Lax form  of scalar type $L_1=0$ and $L_2=0$ (\ref{eq:E6L1L2ad}) is equivalent to the scalar ones in \cite{KNY17,  Nagao16} by using suitable gauge transformations of $y(x)$. On the other hand, the differential $3 \times 3$ matrix Lax form has been given as certain Fuchsian system of differential equations in \cite{Boalch09}, and in \cite{AB06} certain additive Lax form of $2 \times 2$ matrix type has been constructed utilizing moduli spaces of difference connection on $\mathbb{P}^1$ for type $d$-$E_6^{(1)}$, called the difference Painlev\'e {\rm VI} there.\\\\

\noindent
%E6ad
{\bf(e)} Determinant formulae of hypergeometric special solutions

The hypergeometric solutions are constructed as the following explicit forms: 
\begin{equation}\label{eq:E6solad}
\begin{array}{l}
\dfrac{f+a_i}{f+b_j}=\alpha\dfrac{T_{a_i}(\tau_{m,n})T_{a_i}^{-1}(\tau_{m+1,n-1})}
{T_{b_j}^{-1}(\tau_{m,n})T_{b_j}(\tau_{m+1,n-1})} \quad  (i, j=1,2),\quad 
g=-a_2+\beta\dfrac{T_{a_2}(\o{\tau}_{m,n})T_{a_2}^{-1}(\tau_{m+1,n-1})}
{T_{a_1}(\tau_{m,n})T_{a_1}(\o{\tau}_{m+1,n-1})},
\end{array}
\end{equation}
where the determinant $\tau_{m,n}$ is given by
\begin{equation}\label{eq:E6tauad}
\begin{array}l
\ds \tau_{m,n}=\det\Big[(b_1)_i(a_1-j)_j{}_3F_2 \Big(\substack{\displaystyle{b_1+i, b_2,-(m+n)}\\[3mm]{\displaystyle{a_1-j, a_2}}},1\Big)\Big]^n _{i,j=0}, \\[5mm]
\alpha=-\dfrac{(a_i+m+n)(a_i-1)^n(b_j-1)^n}{a_i^{n+1}b_j^n}\dfrac{\prod_{k=1}^2(b_k-a_i)}{\prod_{k=1}^2(a_k-b_j)},\\[5mm]
\beta=-\dfrac{b_1(a_2+m+n)(a_2-1)^n(b_2-a_2)}{a_2^{n+1}(b_1-a_1)}.
\end{array}
\end{equation}
  
These determinant formulae of the generalized hypergeometric solutions (\ref{eq:E6solad}) are given in terms of the terminating hypergeometric series ${}_3F_2$. A certain $1 \times 1$ determinant formula has been constructed in terms of the non-terminating ${}_3F_2$ in \cite{Kajiwara08, KNY17}, and the general determinant formulae have not been given yet in terms of the non-terminating ${}_3F_2$ as far as we know.

%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Case $d$-$D_4^{(1)}$}\label{subsec:D4ad}　\\

%D4ad
\noindent
{\bf(a)} Setting of the Pad\'e interpolation problem on the additive grid\\

In Table (\ref{eq:Ylistad}) the interpolated function and the interpolated sequence are set up as
\begin{equation}\label{eq:D4Yad}
Y(x):=\displaystyle c^x\dfrac{\Gamma(a_1)\Gamma(x+b_1)}{\Gamma(x+a_1)\Gamma(b_1)},\quad Y_s=\displaystyle c^s\dfrac{(b_1)_s}{(a_1)_s},
\end{equation}
and in Table (\ref{eq:Tlistad}) the time evolution is chosen as
\begin{equation}\label{eq:D4Tad}
T: (a_1,b_1,c,m,n) \mapsto (a_1+1,b_1+1,c,m,n).
\end{equation}
\\
%D4ad
\noindent
{\bf(b)} Contiguity three term relations\\

By Definition (\ref{eq:GKHad}) we have the basic quantities
\begin{equation}\label{eq:D4GKHad}
\displaystyle G(x)=c\dfrac{x+b_1}{x+a_1},\quad 
K(x)=\dfrac{a_1(x+b_1)}{b_1(x+a_1)},\quad H(x)=b_1(x+a_1),
\end{equation}
and by the expression (\ref{eq:Drelationad}) we obtain the Casorati determinants
\begin{equation}\label{eq:D4Dad}
\begin{array}{l}
D_1(x)=\sfrac{c_0(x-f){\mathcal N}(x)Y(x)}{G(x)_{\rm den}},\quad
D_2(x)=\sfrac{c_1(x-m-n){\mathcal N}(x)Y(x)}{K(x)_{\rm den}},
\\[5mm]
D_3(x)=\sfrac{c_2{\mathcal N}(x)Y(x)(x+b_1)}{H(x)},
\end{array}
\end{equation}
where $f,c_0,c_1$ and $c_2$ are constants depending on parameters $a_1, b_1, c \in \C^{\times}$ and $m,n \in \Z_{\geq 0}$ but independent of $x$. Then the contiguity relations $L_2=0$ and $L_3=0$ are expressed by
\begin{equation}\label{eq:D4L2L3ad}
\begin{array}{l}
L_2(x)=C_0(f-x)\o{y}(x)-(x-m-n)y(x+1)+\sfrac{(x+b_1)y(x)}{g},\\
L_3(x)=C_1(x-\o{f}-1)y(x)+\sfrac{(x+a_1)\o{y}(x)}{g}
-cx\o{y}(x-1),
\end{array}
\end{equation}
where $C_0=\sfrac{b_1c_0}{c_1}$, $C_1=\sfrac{a_1\o{c}_0}{c_1}$ and $g=\sfrac{c_1}{c_2}$.\\

Take note that in the items (c) and (d) below we study the contiguity relations $L_2=0$ and $L_3=0$ (\ref{eq:D4L2L3ad}) for generic complex parameters $a_0, b_0$ (replacing considering $m, n \in \Z_{\geq 0}$ by $a_0, b_0 \in \C^{\times}$) and generic variables $f, g \in \P^1$ (depending on parameters $a_i, b_i, c \in \C^{\times}, i=0,1$) apart from the Pad\'e interpolation problem (\ref{eq:padead}) with eqs. (\ref{eq:D4Yad}) and (\ref{eq:D4Tad}) (see Remark \ref{rem:genericad}).\\

%D4ad
\noindent
{\bf(c)} The time evolution of the $d$-Painlev\'e equation\\

Compatibility of the contiguity relations $L_2=0$ and $L_3=0$ (\ref{eq:D4L2L3ad}) gives the evolution equations and the constraint on the product $C_0C_1$ as follows:
\begin{equation}\label{eq:D4eqad}
\begin{array}{l}
g\u{g}=\dfrac{c(f+1)(f-a_0-b_0)}{(f+a_1)(f+b_1)},\quad f+\o{f}=\dfrac{a_1+a_0}{cg-1}+\dfrac{b_1+b_0}{g-1}+a_0+b_0-1
\end{array}
\end{equation}
and
\begin{equation}\label{eq:D4C0C1ad}
C_0C_1=\sfrac{(g-1)(cg-1)}{g^2}.
\end{equation}

The evolution equations (\ref{eq:D4eqad}) are equivalent to the $d$-Painlev\'e equation of type $D_4^{(1)}$ given in \cite{Kajiwara08, KNY17, Nagao16, RGTT01, Sakai07}.
%When the coordinates of $f$ and $g$ are changed into $\sfrac{1}{f}$ and $\sfrac{1}{g}$ respectively, 
The 8 singular points in coordinates $(f,g)$ are on the four lines $f=0$, $g=0$, $f=\infty$ and $g=\infty$ as follows:
\begin{equation}\label{eq:D48pad}
\begin{array}{l}
(f,g)=(-a_1,\infty), (-b_1,\infty), (\sfrac{1}{\varepsilon},\sfrac{\{1+(a_0+a_1)\varepsilon\}}{c})_2,\\
\phantom{(f,g)=} (\sfrac{1}{\varepsilon}, 1+(b_0+b_1)\varepsilon)_2, (-1,\infty), (a_0+b_0, \infty).
\end{array}
\end{equation}

Here the third point is a double point at $(\infty,\sfrac{1}{c})$ with the gradient $f(g-\sfrac{1}{c})=\sfrac{a_0+a_1}{c}$ and the fourth point is a double point at $(\infty,1)$ with the gradient $f(g-1)=b_0+b_1$. (The meaning of the two double points is also written in \cite{KNY17}.)\\

%D4ad
\noindent
{\bf(d)} The additive difference Lax form of scalar type\\

The contiguity relations $L_2=0$ and $L_3=0$ (\ref{eq:D4L2L3ad}) give two scalar additive Lax equations $L_1=0$ and $L_2=0$ expressed by
\begin{equation}\label{eq:D4L1L2ad}
\begin{array}{l}
L_1(x)=(g-1)(c g-1)\Big[\dfrac{c(a_1+a_0)}{c g-1}+\dfrac{b_1+b_0}{g-1}-\dfrac{x+f+a_1+b_1}{g}\Big]y(x)
\\[5mm]\phantom{L_1(x)}
-\dfrac{cx(x+b_1-1)}{x-f-1}\Big[y(x-1)-\dfrac{g(x-a_0-b_0-1)}{x+b_1-1}y(x)\Big]
\\[5mm]\phantom{L_1(x)}
-\dfrac{(x-a_0-b_0)(x+a_1)}{x-f}\Big[y(x+1)-\dfrac{x+b_1}{g(x-a_0-b_0)}y(x)\Big],
\\[5mm]
L_2(x)=(f-x)\o{y}(x)-(x-a_0-b_0)y(x+1)+\sfrac{(x+b_1)y(x)}{g}.
\end{array}
\end{equation}

The additive Lax form  of scalar type  (\ref{eq:D4L1L2ad}) is equivalent to the scalar ones in \cite{KNY17,  Nagao16} by using suitable gauge transformations of $y(x)$. On the other hand, in \cite{AB06} certain additive Lax form of 2 $\times$ 2 matrix type has been obtained utilizing moduli spaces of difference connection on $\mathbb{P}^1$ for type $d$-$D_4^{(1)}$, called the difference Painlev\'e {\rm V}. Concerning the differential Lax form for type $d$-$D_4^{(1)}$, the $2 \times 2$ matrix Lax pair and the scalar one have been derived respectively in \cite{GORS98} and \cite{KNY17} by using a Schlesinger transformation of differential equations.\\

\noindent
%D4ad
{\bf(e)} Determinant formulae of hypergeometric special solutions　\\

The hypergeometric solutions are constructed as the following explicit forms: 
\begin{equation}\label{eq:D4solad}
\begin{array}{l}
\dfrac{f+a_1}{f+b_1}=\alpha\dfrac{T_{a_1}(\tau_{m,n})T_{a_1}^{-1}(\tau_{m+1,n-1})}
{T_{b_1}^{-1}(\tau_{m,n})T_{b_1}(\tau_{m+1,n-1})},\quad 
\dfrac{1}{g}=1-\dfrac{b_1(c-1)}{b_1-a_1}
\dfrac{\o{\tau}_{m,n}\tau_{m+1,n-1}}
{T_{a_1}(\tau_{m,n})T_{a_1}(\o{\tau}_{m+1,n-1})},
\end{array}
\end{equation}
where the determinant $\tau_{m,n}$ is given by
\begin{equation}\label{eq:D4tauad}
\begin{array}l
\ds \tau_{m,n}=\det\Big[(b_1)_i(a_1-j)_j{}_2F_1 \Big(\substack{\displaystyle{b_1+i,-(m+n)}\\[3mm]{\displaystyle{a_1-j}}},c\Big)\Big]^n _{i,j=0}, \quad 
\alpha=c\dfrac{(a_1+m+n)(a_1-1)^n(b_1-1)^n}{a_1^{n+1}b_1^n}.
\end{array}
\end{equation}
  
These determinant formulae of the Gauss hypergeometric solutions (\ref{eq:D4solad}) are given in terms of the terminating hypergeometric series ${}_2F_1$. A certain $1 \times 1$ determinant formula has been constructed in terms of the non-terminating ${}_2F_1$ (e.g. \cite{KNY17}), and the general determinant formulae have not  been given yet in terms of the non-terminating ${}_2F_1$ as far as we know.
 
%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Case $d$-$A_3^{(1)}$}\label{subsec:A3ad}　\\

%A3ad
\noindent
{\bf(a)} Setting of the Pad\'e interpolation problem on the additive grid\\

In Table (\ref{eq:Ylistad}) the interpolated function and the interpolated sequence are set up as
\begin{equation}\label{eq:A3Yad}
Y(x):=d^x\dfrac{\Gamma(x+b_1)}{\Gamma(b_1)},\quad Y_s=d^s(b_1)_s,
\end{equation}
and in Table (\ref{eq:Tlistad}) the time evolution is chosen as
\begin{equation}\label{eq:A3Tad}
T: (b_1,d,m,n) \mapsto (b_1+1,d,m,n).
\end{equation}\\

%A3ad
\noindent
{\bf(b)} Contiguity three term relations\\

By Definition (\ref{eq:GKHad}) we have the basic quantities
\begin{equation}\label{eq:A3GKHad}
\displaystyle G(x)=d(x+b_1), \quad 
K(x)=\dfrac{x+b_1}{b_1},\quad H(x)=b_1,
\end{equation}
and by the expression (\ref{eq:Drelationad}) we obtain the Casorati determinants
\begin{equation}\label{eq:A3Dad}
\begin{array}{l}
D_1(x)=\sfrac{c_0(x-f){\mathcal N}(x)Y(x)}{G(x)_{\rm den}},\quad
D_2(x)=\sfrac{c_1(x-m-n){\mathcal N}(x)Y(x)}{K(x)_{\rm den}},
\\[5mm]
D_3(x)=\sfrac{c_2{\mathcal N}(x)Y(x)(x+b_1)}{H(x)},
\end{array}
\end{equation}
where $f,c_0,c_1$ and $c_2$ are constants depending on parameters $b_1, d \in \C^{\times}$ and $m,n \in \Z_{\geq 0}$ but independent of $x$. Then the contiguity relations $L_2=0$ and $L_3=0$ are expressed by
\begin{equation}\label{eq:A3L2L3ad}
\begin{array}{l}
L_2(x)=C_0(f-x)\o{y}(x)-(x-m-n)y(x+1)+\sfrac{(x+b_1)y(x)}{g},\\
L_3(x)=C_1(x-\o{f}-1)y(x)+\sfrac{\o{y}(x)}{g}-dx\o{y}(x-1),
\end{array}
\end{equation}
where $C_0=\sfrac{b_1c_0}{c_1}$, $C_1=\sfrac{\o{c}_0}{c_1}$ and $g=\sfrac{c_1}{c_2}$.\\

Take note that in the items (c) and (d) below we study the contiguity relations $L_2=0$ and $L_3=0$ (\ref{eq:A3L2L3ad}) for generic complex parameters $a_0, b_0$ (replacing considering $m, n \in \Z_{\geq 0}$ by $a_0, b_0 \in \C^{\times}$) and generic variables $f, g \in \P^1$ (depending on parameters $a_0, b_0, b_1, d \in \C^{\times}$) apart from the Pad\'e interpolation problem (\ref{eq:padead}) with eqs. (\ref{eq:A3Yad}) and (\ref{eq:A3Tad}) (seeRemark \ref{rem:genericad})\\

%A3ad
\noindent
{\bf(c)} The time evolution of the $d$-Painlev\'e equation\\

Compatibility of the contiguity relations $L_2=0$ and $L_3=0$ (\ref{eq:A3L2L3ad}) gives the evolution equations and the constraint on the product $C_0C_1$ as follows:
\begin{equation}\label{eq:A3eqad}
\begin{array}{l}
g\u{g}=\dfrac{f+b_1}{d(f+1)(f-a_0-b_0)},\quad f+\o{f}=\dfrac{1}{dg}+\dfrac{b_0+b_1}{g-1}+a_0+b_0-1
\end{array}
\end{equation}
and
\begin{equation}\label{eq:A3C0C1ad}
C_0C_1=\sfrac{d(g-1)}{g}.
\end{equation}

The evolution equations (\ref{eq:A3eqad}) are equivalent to the $d$-Painlev\'e equation of type $A_3^{(1)}$ given in \cite{Kajiwara08, KNY17, Nagao16, RGTT01, Sakai07}.
%When the coordinates of $f$ and $g$ are changed into $\sfrac{1}{f}$ and $\sfrac{1}{g}$ respectively, 
The 8 singular points in coordinates $(f,g)$ are on the four lines $f=0$, $g=0$, $f=\infty$ and $g=\infty$ as follows:
\begin{equation}\label{eq:A38pad}
\begin{array}{l}
(f,g)=(-1,0), (a_0+b_0, 0), (\sfrac{1}{\varepsilon},\sfrac{\varepsilon}{d}(1+m\varepsilon))_3, (\sfrac{1}{\varepsilon}, 1+(b_0+b_1)\varepsilon)_2, (-b_1, \infty).
\end{array}
\end{equation}

Here the third point is a triple point at $(\infty,0)$ and the fourth point is a double point at $(\infty,1)$ with the gradient $f(g-1)=b_0+b_1$. (The meaning of the triple and double points is also written in \cite{KNY17}.)\\

%A3ad
\noindent
{\bf(d)} The additive difference Lax form of scalar type\\

The contiguity relations $L_2=0$ and $L_3=0$ (\ref{eq:A3L2L3ad}) give two scalar additive Lax equations $L_1=0$ and $L_2=0$ expressed by
\begin{equation}\label{eq:A3L1L2ad}
\begin{array}{l}
L_1(x)=(g-1)\Big[\dfrac{1}{dg}+\dfrac{b_1+b_0}{g-1}-(x+f-a_0-b_0)\Big]y(x)
\\[5mm]\phantom{L_1(x)}
-\dfrac{x(x+b_1-1)}{x-f-1}\Big[y(x-1)-\dfrac{g(x-a_0-b_0-1)}{x+b_1-1}y(x)\Big]
\\[5mm]\phantom{L_1(x)}
-\dfrac{x-a_0-b_0}{d(x-f)}\Big[y(x+1)-\dfrac{x+b_1}{g(x-a_0-b_0)}y(x)\Big],
\\[5mm]
L_2(x)=(f-x)\o{y}(x)-(x-a_0-b_0)y(x+1)+\sfrac{(x+b_1)y(x)}{g}.\\
\end{array}
\end{equation}

The additive Lax form of scalar type $L_1=0$ and $L_2=0$ (\ref{eq:A3L1L2ad}) is equivalent to the scalar ones in \cite{KNY17,  Nagao16} by using suitable gauge transformations of $y(x)$. Concerning the differential Lax form for type $d$-$A_3^{(1)}$, the $2 \times 2$ matrix Lax pair and the scalar one have been derived respectively in \cite{GORS98} and \cite{KNY17} by using a Schlesinger transformation of differential equations.\\

\noindent
%A3ad
{\bf(e)} Determinant formulae of hypergeometric special solutions

The hypergeometric solutions are constructed as the following explicit forms: 
\begin{equation}\label{eq:A3solad}
\begin{array}{l}
f=-b_1+\dfrac{db_1^n}{d(b_1-1)^n}\dfrac{T_{b_1}^{-1}(\tau_{m,n})T_{b_1}(\tau_{m+1,n-1})}{\tau_{m,n}\tau_{m+1,n-1}}
,\quad 
g=1-db_1\dfrac{\o{\tau}_{m,n}\tau_{m+1,n-1}}
{\tau_{m,n}\o{\tau}_{m+1,n-1})},
\end{array}
\end{equation}
where the determinant $\tau_{m,n}$ is given by
\begin{equation}\label{eq:A3tauad}
\begin{array}l
\ds \tau_{m,n}=\det\Big[(b_1)_i(-(m+n))_j {}_2F_0 \Big(\substack{\displaystyle{b_1+i,-(m+n)+j}\\[3mm]{\displaystyle{0}}},d\Big)\Big]^n _{i,j=0}.
\end{array}
\end{equation}
  
These determinant formulae of the Kummer hypergeometric solutions (\ref{eq:A3solad}) are given in terms of the terminating hypergeometric series ${}_2F_0$. A certain $1 \times 1$ determinant formula has been constructed in terms of the non-terminating ${}_1F_1$ (e.g. \cite{KNY17}), and the general determinant formulae have not been given yet in terms of the non-terminating ${}_1F_1$ (or ${}_2F_0$) as far as we know.
%%%%%%%%%%%%%%%%%%%%%%%%

\section{Conclusions}\label{sec:conc}
\subsection{Summary}　\\

\noindent
In this paper for the interpolated function $Y(x)$ and the interpolated sequence $Y_s$ given in Table (\ref{eq:Ylistad}), we set up the Pad\'e interpolation problem on the additive gird, related to the $d$-Painlev\'e equations of type $E_7^{(1)}$, $E_6^{(1)}$, $D_4^{(1)}$ and $A_3^{(1)}$. Then for the time evolution $T$ given in Table (\ref{eq:Tlistad}), we set up another Pad\'e interpolation problem on the additive gird. By solving these problems, we derived the evolution equations, the Lax pairs of scalar type and the determinant formulae of the special solutions for the corresponding $d$-Painlev\'e equations. The detailed results were given in Section \ref{sec:detail results}. 

\subsection{Problems}　\\

\noindent
Some open problems related to the results of this paper are as follows:

{\bf 1.} One may be interested in studying whether the Pad\'e interpolation method on the additive grid can be applied to the additive difference ($d$-) Painlev\'e equations of type $E_8^{(1)}$. 
%and especially it may be interesting that determinant formulae are constructed in terms of terminating generalized hypergeometric special solution for type $d$-$E_8^{(1)}$.

{\bf 2.} Differently from the additive grid, it may be interesting to investigate whether the Pad\'e method on the differential grid (i.e. Pad\'e approximation) can be also applied to the $d$-Painlev\'e equations of type $D_4^{(1)}$, $A_3^{(1)}$ and $A_2^{(1)}$ by using a Schlesinger transformation of differential equations.

{\bf 3.} It may be interesting to study whether the Pad\'e method can be further applied to other generalized $d$-Painlev\'e systems, for example an additive difference analogues of the Garnier system and a higher order Painlev\'e system, which are called ``{\it $d$-Garnier system}" \cite{DST13, DT14, OR16-1} and ``{\it higher order $d$-Painlev\'e system}'' respectively.\\

%%%%%%%%%%%
\appendix
\section{Sufficiency for the compatibility of the Lax pair}\label{sec:Proof}

%\subsection{Proof of the compatibility condition of the Lax pair}\label{subsec:proof}　\\
%This subsection is based on the energetic discussion with Y.Yamada. 
This appendix is the same as in \cite{Nagao16}. In Section \ref{subsec:E7ad} we gave the $d$-$E_7^{(1)}$ equation (\ref{eq:E7eqad}) as the necessary condition for the compatibility of the Lax pair (\ref{eq:E7L1L2ad}). In this appendix, we prove that the $d$-$E_7^{(1)}$ equation is the sufficient condition for the compatibility of the Lax pair. 

Eliminating $\o{y}(x)$ and $\o{y}(x-1)$ from $L_2(x)= L_2(x-1)=L_3(x)=0$ (\ref{eq:E7L2L3ad}), one constructs the linear equation $L_1=0$ among 
$y(x+1)$, $y(x)$ among $y(x-1)$ where
\begin{equation}\label{eq:E7L1ad}
\begin{array}l
\ds L_1(x)= \frac{(x-a_0-b_0)\prod_{i=1}^3(x+a_i)}{x-f}y(x+1)+\frac{x\prod_{i=1}^3(x+b_i-1)}{(x-f-1)}y(x-1)\\[5mm]
\ds\phantom{L_1(x): }-\frac{1}{x-h}\Big[\frac{A_2(x)(x-g)}{x-f}+\frac{V(x-1)}{(x-f-1)(x-g-1)}\Big]y(x)
\end{array}
\end{equation}
and
\begin{equation}\label{eq:E7Vad}
V(x)=(x-h)(x-h+1)A_1(x)-C_0C_1(x-f)(x-\o{f}).
\end{equation}
Here the variable $\o{f}$ and the product $C_0C_1$ in (\ref{eq:E7Vad}) should be viewed as functions in terms of $f$ and $g$, and they are determined  in (\ref{eq:E7eqad}) and (\ref{eq:E7C0C1ad}) respectively. The expression $L_1$ (\ref{eq:E7L1ad}) is rewritten into (\ref{eq:E7L1L2ad}) by using (\ref{eq:E7eqad}) and (\ref{eq:E7C0C1ad}). 
 
Similarly, by eliminating $y(x)$ and $y(x+1)$ from $L_2(x)= L_3(x)=L_3(x+1)=0$ (\ref{eq:E7L2L3ad}), we obtain the linear equation $L_1^*=0$ among
$\o{y}(x+1)$, $\o{y}(x)$ and $\o{y}(x-1)$ where
\begin{equation}\label{eq:E7L1*ad}
\begin{array}l
\ds L_1^*(x)= \frac{(x-a_0-b_0)(x+a_1+1)(x+a_2)(x+a_3+1)}{x-\o{f}}\o{y}(x+1)
\\[5mm]\phantom{L_1^*(x)}
\ds+\frac{x(x+b_1)(x+b_2-1)(x+b_3)}{(x-\o{f}-1)}\o{y}(x-1)
%\\[5mm]\ds\phantom{L_1^*(x): }
\ds-\frac{1}{x-h}\Big[\frac{A_2(x)(x-g-1)}{x-\o{f}-1}+\frac{V(x)}{(x-\o{f})(x-g)}\Big]y(x).
\end{array}
\end{equation}
The sufficiency for the compatibility means that $T(L_1(x))\propto T_1^*(x)$ holds when the $d$-$E_7^{(1)}$ equation (\ref{eq:E7eqad}) is satisfied. In order to prove the sufficiency, we characterize $L_1$ and $L_1^*$ as polynomials in terms of $x$, and compare these characterization.
%\begin{figure}[http] 
\begin{center}\setlength{\unitlength}{1.2mm}
\begin{equation}\label{eq:compatibility}
\begin{picture}(50,28)(-5,-3)
\put(-1,23){$\o{y}(x-1)$}
\put(20,23){$\o{y}(x)$}
\put(41,23){$\o{y}(x+1)$}
\put(-1,-4){$y(x-1)$}
\put(20,-4){$y(x)$}
\put(41,-4){$y(x+1)$}
\put(1,4){$L_2(x-1)$}
\put(26,4){$L_2(x)$}
\put(11,14){$L_3(x)$}
\put(29,14){$L_3(x+1)$}
%\put(11,-4){$L_1(x)$}
%\put(0,-1){\line(1,0){43}}
\put(0,0){\line(1,0){20}}
\put(0,0){\line(0,1){20}}
\put(0,20){\line(1,-1){20}}
\put(1,21){\line(1,0){20}}
\put(21,1){\line(0,1){20}}
\put(1,21){\line(1,-1){20}}
\put(22,0){\line(1,0){20}}
\put(22,0){\line(0,1){20}}
\put(22,20){\line(1,-1){20}}
\put(23,21){\line(1,-1){20}}
\put(23,21){\line(1,0){20}}
\put(43,21){\line(0,-1){20}}
%\put(0,22){\line(1,0){43}}
\end{picture}
\end{equation}
\end{center}

%The equation $L_1$ (\ref{eq:E7L1}) has the following characterization.
\begin{lem}\label{lem:E7L1p}
The expression $(x-f)(x-f-1)L_1(x)$ (\ref{eq:E7L1ad}) (or (\ref{eq:E7L1L2ad})) has the following characterization:\\
\hspace{5mm}(i) It is a linear equation among $y(x+1)$, $y(x)$ and $y(x-1)$, and the coefficients of these terms are polynomials of degree $5$ in $x$.\\
\hspace{5mm}(ii) The coefficients of $y(x+1)$ (resp. $y(x-1)$) has zero at $x=-a_1$,$-a_2$,$-a_3$, $a_0+b_0$ (resp. $x=-b_1+1$,$-b_2+1$, $-b_3+1$, $0$).\\ 
\hspace{5mm}(iii) Under the condition 
\begin{equation}\label{eq:cond}
\begin{array}l
\displaystyle\frac{y(x+1)}{y(x)}=1+\frac{a_0}{x}+\frac{a_0(a_0-1)/2}{x^2}+\frac{w}{x^3}+O\Big(\frac{1}{x^4}\Big), \\[5mm]
\displaystyle \frac{y(x-1)}{y(x)}=1-\frac{a_0}{x}+\frac{a_0(a_0-1)/2}{x^2}-\frac{w}{x^3}+O\Big(\frac{1}{x^4}\Big),
\end{array}
\end{equation}
the terms $x^5, \ldots, x^2$ in the expression $(x-f)(x-f-1)L_1(x)$ vanish, namely $(x-f)(x-f-1)L_1(x)=O(x^1)$ around $x=\infty$. Here $w \in \C$ is an arbitrary constant.\\
\hspace{5mm}(iv) The equation $(x-f)(x-f-1)L_1=0$ holds at the two points $x=f, f+1$ where
\begin{equation}\label{eq:y}
\dfrac{y(f+1)}{y(f)} =\dfrac{(f+b_1)(f+b_3)(f-g)}{(f+a_2)(f-a_0-b_0)(f-h)}.
% \quad \mbox{for}\quad x=f, f-1.
\end{equation}
Conversely, the expression $(x-f)(x-f-1)L_1(x)$ is uniquely characterized by these properties $(i)-(iv)$.
\sq
\end{lem}

\prf
%\begin{proof}
The property (i) is obtained by the relations (\ref{eq:E7C0C1ad}). Concretely, the expression $\frac{V(x-1)}{x-g-1}$ reduces to a polynomial of degree 5 in $x$ under the first relation of (\ref{eq:E7C0C1ad}). Moreover, the coefficient of the term $y(x)$ is obtained as a polynomial of degree 5 in $x$ by using the second relation of (\ref{eq:E7C0C1ad}). The property (ii) is trivial. The property (iii) can easily be checked by the condition (\ref{eq:cond}). The property (iv) follows by substituting $x=f, f+1$ into the equation $L_1(x)=0$. 
\qed
%\end{proof}

We remark that two points $x=f, f+1$ are apparent singularities in the sense that at those two points the equation $(x-f)(x-f-1)L_1(x)=0$ is satisfied by the same condition (in this case (\ref{eq:y})).

The following Lemma (and its proof) is similar to Lemma \ref{lem:E7L1p}.
%The linear equation (\ref{eq:E7L4}) has the following characterization.
\begin{lem}\label{lem:E7L1*p}
The expression $(x-\o{f})(x-\o{f}-1)L_1^*(x)$ (\ref{eq:E7L1*ad}) has the following characterization:\\
\hspace{5mm}(i) It is a linear three term expression among $\o{y}(x+1)$ and $\o{y}(x)$ and $\o{y}(x-1)$, and the coefficients of these terms are polynomials of degree $5$ in $x$.\\
\hspace{5mm}(ii) The coefficients of $\o{y}(x+1)$ (resp. $\o{y}(x-1)$) has zero at $x=-a_1-1$,$-a_2$,$-a_3-1$, $a_0+b_0$ (resp. $x=-b_1$,$-b_2+1$, $-b_3$, $0$).\\ 
\hspace{5mm}(iii) Under the condition 
\begin{equation}\label{eq:cond*}
\begin{array}l
\displaystyle\frac{\o{y}(x+1)}{\o{y}(x)}=1+\frac{a_0}{x}+\frac{\o{a}_0(a_0-1)/2}{x^2}+\frac{\o{w}}{x^3}+O\Big(\frac{1}{x^4}\Big), \\[5mm]
\displaystyle \frac{\o{y}(x-1)}{\o{y}(x)}=1-\frac{a_0}{x}+\frac{a_0(a_0-1)/2}{x^2}-\frac{\o{w}}{x^3}+O\Big(\frac{1}{x^4}\Big),
\end{array}
\end{equation}
the terms $x^5, \ldots, x^2$ in the expression $(x-\o{f})(x-\o{f}-1)L_1^*(x)$ vanish, namely $(x-\o{f})(x-\o{f}-1)L_1^*(x)=O(x^1)$ around $x=\infty$. Here $w \in \C$ is the same  arbitrary constant in (\ref{eq:cond}).\\
\hspace{5mm}(iv) The equation $(x-\o{f})(x-\o{f}-1)L_1^*=0$ holds at the two points $x=\o{f}, \o{f}+1$ where
\begin{equation}\label{eq:yu}
\dfrac{\o{y}(\o{f}+1)}{\o{y}(\o{f})} =\dfrac{(\o{f}+b_2)(\o{f}+1)(\o{f}+h-1)}{(\o{f}+a_1+1)(\o{f}+a_3+1)(\o{f}-g)}.
% \quad \mbox{for}\quad x=f, f-1.
\end{equation}
Conversely, the expression $(x-\o{f})(x-\o{f}-1)L_1^*(x)$ is uniquely characterized by these properties $(i)-(iv)$.
\sq
\end{lem}
%\begin{proof}
%This proof is similar to that of Lemma \ref{lem:E7L1p}.
%\end{proof}

The following is the main result of this paper.
\begin{prop}\label{prop:Lax}
The linear equations $L_1=0$ and $L_2=0$ (\ref{eq:E7L1L2ad}) for the  unknown function $y(x)$ are compatible
if and only if the $d$-$E_7^{(1)}$ equation (\ref{eq:E7eqad}) is satisfied. 
\sq
\end{prop}

\prf
%\begin{proof}
The compatibility means that the shift operator $T$ changes the equation $L_1=0$ into the equation $L_1^{\ast}=0$, i.e. the commutativity of the following: 
\begin{equation}\label{eq:classification}\nonumber
\begin{array}{cccc}
L_1^*=0 \ (\mbox{Lemma} \  \ref{lem:E7L1*p})\quad &\Leftrightarrow&L_1^*=0\  (\ref{eq:E7L1*ad})\\
\uparrow&&\uparrow\\
\mbox{$T$-shift}\hspace{2mm}(\ref{eq:E7Tad})&&L_2= L_3=0 \hspace{2mm}(\ref{eq:E7L2L3ad})\\
\uparrow&&\downarrow\\
L_1=0 \ (\mbox{Lemma} \  \ref{lem:E7L1p})\ &\Leftrightarrow& L_1=0 \ (\ref{eq:E7L1ad}) &\Leftrightarrow L_1=0 \ (\ref{eq:E7L1L2ad}).
\end{array}
\end{equation}
This commutativity is almost clear from the characterizations (i),(ii) of the equation $L_1=0$ (respectively $L_1^*=0$) in Lemma \ref{lem:E7L1p} (respectively Lemma \ref{lem:E7L1*p}).
The remaining task is to check that the operator $T$ changes expression (\ref{eq:y}) into expression (\ref{eq:yu}), utilizing the characterization (iii) of the equation $L_1=0$ (respectively $L_1^*=0$) and the first of equation (\ref{eq:E7eqad}).
\qed
%\end{proof}

As the point of the proof, the following two are applied to type $d$-$E_7^{(1)}$ together: The first is that the equation $L_1(f,\o{f}, g)=0$ in terms of $f$, $\o{f}$ and $g$ is derived from the equations $L_2(f,g)=0$ and $L_3(\o{f},g)=0$ (see \cite{KNY17,NTY13,Yamada09-2,Yamada11}). The second is that the equation $L_1(f,\o{f}, g)=0$ is characterized as a polynomial in terms of $x$ (see \cite{Nagao16,NY16}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgment}

\noindent
The author is grateful to Professor Yasuhiko Yamada for valuable discussions on this research, and for his encouragement. The author also thanks Professors Kenji Kajiwara 
%and the referee 
for stimulating comments and for kindhearted support. This work was partially supported by Expenses Revitalizing Education and Research of Akashi College.

%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{A}
%\bibitem{Andrews72}
%Andrews G.E., {\it Summation and transformation for basic Appell series}, J. London Math. Soc. {\bf 4} (2) (1972),  618--622.
%\bibitem{Andrews75}
%Andrews G.E., {\it Problems and prospectives for basic hypergeometric functions, in: Theory and application of special functions}, Proc. Advanced Sem. Math. Res. Center, University of Wisconsin, Madison, WI, (1975), 19--224.
\bibitem{AB06}
Arinkin D., and Borodin A., {\it Moduli spaces of d-connections and difference Painlev\'e equations}
Duke Mathematics Journal {\bf 134}, no. 2 (2006): 515--56.
\bibitem{Bailey35}
Bailey W., N., {\it Generalized hypergeometric series}, Cambridge tracts in mathematics and mathematical physics No. 32, Cambridge, Cambridge University Press, (1935).
%\bibitem{Boalch06}
%Boalch P., {\it The fifty-two icosahedral solutions to Painlev\'e {\rm VI}}. J. Reine Angew. Math. {\bf 596} (2006), 183--214.
\bibitem{Boalch09}
Boalch P., {\it Quivers and difference Painlev\'e equations} Groups and Symmetries 25--51, CRM Proceedings in Lecture Notes {\bf 47}, Providence, RI: American Mathematical Society, (2009).
\bibitem{Clarkson13}
Clarkson P.A., {\it Recurrence coefficients for discrete orthonormal polynomials and the Painlev\'e equations}, J. Phys., A {\bf 46}, no.18, (2013), 185205--185222.
\bibitem{Conte99}
Conte R. (Ed.), {\it The Painlev\'e Property--One Century Later}, CRM Series in Mathematical Physics, Springer, Berlin, 1999.
%\bibitem{DM00}
%Dubrovin B., and Mazzocco M., {\it Monodromy of certain Painlev\'e {\rm VI} transcendents and reflection groups}, Invent. Math. {\bf 141} (2000) 55--147.
\bibitem{DST13}
Dzhamay A., Sakai H., and Takenawa T., {\it Discrete Hamiltonian Structure of Schlesinger Transformations}, arXiv:1302.2972 [math-ph].
\bibitem{DT14}
Dzhamay A., and Takenawa T., {\it Geometric Analysis of Reductions from Schlesinger transformations to difference Painlev\'e equations}, arXiv:1408.3778 [math-ph].
\bibitem{Forrester10} Forrester P. J., {\it Log-Gases and Random Matrices}, London Mathematical Society Monographs 
{\bf 34} (Princeton: Princeton University Press, 2010).
\bibitem{Fuchs1907}
Fuchs R., \``{U}ber lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegene wesentlich singul\``{a}ren Stellen. Math. Ann. 63, (1907) 301--321
%\bibitem{Gambier1910}
%Gambier B., {\it Sur les \'{e}quations diff\'{e}retielles du second ordre et du premier degr\'{e} dont l’int\'{e}grale g\'{e}n\'{e}rale est \'{a} points critiques fixes.} Acta. Math. {\bf33}, (1910) 1--55
%\bibitem{Garnier12}
%Garnier R., {\it Sur des \'{e}quations diff\'{e}rentielles du troisi\'{e}me ordre dont l’int\'{e}grale g\'{e}n\'{e}rale est uniforme et sur une classe d’\'{e}quations nouvelles d’ordre sup\'{e}rieur dont l’int\'{e}grale g\'{e}n\'{e}rale a ses points critiques fixes.} Ann. Sci. Ecole Norm. Super. {\bf29}, (1912) 1--126
%\bibitem{GaR90}
%Gasper G., and Rahman M., {\it Basic Hypergeometric Series}, Encyclopedia of Mathematics
%and Its Applications, Volume {\bf 35} Cambridge University Press, Cambridge, (1990).
\bibitem{GaR04}
Gasper G., and Rahman M., {\it Basic Hypergeometric Series. With a foreword by Richard Askey. Second edition. Encyclopedia of Mathematics and its Applications}, Volume {\bf 96}, Cambridge University Press, Cambridge, (2004). 
\bibitem{GORS98}
Grammaticos B., Ohta Y., Ramani A., and Sakai H., {\it Degeneration through coalescence of the $q$-Painlev\'e {\rm VI} equations}
J. Phys. A : Math. Gen. {\bf 31}(15), 3545--3558, (1998) 
%\bibitem{GNPRS94}
%Grammaticos B., Nijhoff F.W., Papageorgiou V.G., Ramani A., and Satsuma J., {\it Linearization and solution of the discrete Painlev\'e III equation}, Phys. Lett. A {\bf185}, (1994), 446.
%\bibitem{GNR99}
%Grammaticos B., Nijhoff F.W., and Ramani A., {\it Discrete Painlev\'e equations, The Painlev\'e Property: One Century Later}, Editor R. Conte, CRM Ser. Math. Phys., Springer, New York, (1999), 413--516.
\bibitem{GR99}
Grammaticos B., and Ramani A., {\it On a novel $q$-discrete analogue of the Painlev\'{e} \rm{VI} equation}, Phys. Lett., A{\bf257} (1999), 288--292. 
%\bibitem{GR04}
%Grammaticos B., and A. Ramani, {\it Discrete Painlev\'e equations: a review}, in: Grammaticos B., Kosmann-Schwarzbach Y., Tamizhmani T. (Eds.), Discrete Integrable Systems, Lecture Notes in Physics, vol. {\bf 644}, Springer, Berlin, 2004, pp. 245--321.
%\bibitem{GRO03}
%Grammaticos B., Ramani A., Ohta Y., {\it A Unified Description of the Asymmetric $q$-$P_{V}$ and $d$-$P_{IV}$ Equations and their Schlesinger Transformations}. J. Nonlinear Math. Phys. {\bf10}(2), 215--228 (2003)
%\bibitem{GRP91}
%Grammaticos B., Ramani A., and Papageorgiou V. G., {\it Do Integrable Mappings Have the Painlev\'e Property?} Phys. Rev. Lett. {\bf67} 1825--18258 (1991) 
%\bibitem{GL82}
%Gromak V.I. and Lukashevich N.A., {\it Special class of solutions of Painlev\'e equations}, Differential Equations {\bf 18 }(1982) 317--326.
%\bibitem{GLS02}
%Gromak V.I., Laine I. and Shimomura S., {\it Painlev\'e Differential Equations in the Complex Plane}, de Gruyter Studies in Mathematics, vol. {\bf 28}, Walter de Gruyter \& Co., Berlin, 2002.
%\bibitem{HK07}
%Hamamoto T., and Kajiwara K., {\it Hypergeometric solutions to the q-Painlev\'e equation of type $A_4^{(1)}$}, J. Phys. A: Math. Theor., {\bf 40} (2007), 12509--12524. 
\bibitem{Ikawa13}
Ikawa Y., {\it Hypergeometric Solutions for the $q$-Painlev\'e Equation of Type $E_6^{(1)}$ by the Pad\'e method}, Lett. Math. Phys., Volume {\bf 103}, Issue 7  (2013), 743--763.
\bibitem{IKSY91}
Iwasaki K., Kimura H., Shimomura S. and Yoshida M., {\it From Gauss to Painlev\'e--A Modern Theory of Special Functions}, Aspects of Mathematics, vol. {\bf E16}, Vieweg, 1991.
\bibitem{Jacobi}
Jacobi,C.G.J., {\it \``{U}ber die Darstellung einer Reihe gegebner Werthe durch eine gebrochne rationale Function}, J. Reine Angew. Math., {\bf 30} (1846) 127--156.
%\bibitem{JM81-1}
%Jimbo M., Miwa T., and Ueno K., {\it Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I.} Physica {\bf2D} (1981) 306--52
\bibitem{JM81-2}
Jimbo M., and Miwa T., {\it Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II.} Physica {\bf2D} (1981) 407-448
%\bibitem{JM81-3}
%Jimbo M., and Miwa T., {\it Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. III.}Physica {\bf4D} (1981) 26--46
%\bibitem{JS96}
%Jimbo M., and Sakai H., {\it A $q$-analog of the sixth Painlev\'e equation}, Lett. Math. Phys., {\bf 38} (1996), 145--154.
%\bibitem{Kajihara04}
%Kajihara Y., {\it Euler transformation formula for multiple basic hypergeometric series of type $A$ and some applications}, Advances in Mathematics {\bf187} Issue 1, (2004), 53--97.
\bibitem{Kajiwara08}
Kajiwara K., {\it Hypergeometric solutions to 
the additive discrete Painlev\'e equations with affine Weyl group symmetry of type E}, Reports of Research Institute for Applied Mechanics Symposium, No.19ME{S2} (2008), No.3.
%\bibitem{KK03}
%Kajiwara K., and Kimura K., {\it On a q-difference Painlev\'e III equation, I.Derivation, symmetry and Riccati type solutions.} J. Nonlinear Math. Phys.{\bf 10} (2003) 86--102.
%\bibitem{Kitaev06}
%Kitaev A.V., {\it Remarks towards a classification of $RS^2_4(3)$-transformations and algebraic solutions of the sixth Painlev\'e equation. Th\'eories asymptotiques et \'equations de Painlev\'e}, 199--227, S\'emin. Congr., {\bf 14}, Soc. Math. France, Paris, 2006.
%\bibitem{KLM94}
%Kitaev A.V., Law C.K., and McLeod J.B.,
%{\it Rational solutions of the fifth Painlev\'e equation}, Differential Integral Equations, {\bf 7} (1994), pp. 967--1000
%\bibitem{KMNOY03}
%Kajiwara K., Masuda T., Noumi M., Ohta Y., and Yamada Y., ${}_{10}E_{9}$ solution to the elliptic Painlev\'e equation, J. Phys. A: Math. Gen. {\bf 36} (2003), L263-L272, nlin.SI/0303032.
%\bibitem{KMNOY04}
%Kajiwara K., Masuda T., Noumi M., Ohta Y., and Yamada Y.,
%{\it Hypergeometric solutions to the $q$-Painlev\'e equations}, 
%Int. Math. Res. Not. 2004, {\bf 47} (2004), 2497--2521.
%\bibitem{KMNOY05}
%Kajiwara K., Masuda T., Noumi M., Ohta Y., and Yamada Y., {\it Construction of hypergeometric solutions to the $q$-Painlev\'e equations}, Int. Math. Res. Not. (2005) 1439--1463
%\bibitem{KMNOY06}
%Kajiwara K., Masuda T., Noumi M., Ohta Y., and Yamada Y., {\it Point configurations, Cremona transformations and the elliptic difference Painlev\'e equation}. (English, French summary) Th\'eories asymptotiques et \'equations de Painlev\'e, 169--198,
%S\'emin. Congr., {\bf 14}, Soc. Math. France, Paris, 2006. 
%\bibitem{KNY01}
%Kajiwara K., Noumi M., and Yamada Y., {\it A study on the fourth $q$-Painlev\'e equation}, J. Phys. A: Math. Gen., {\bf 34} (2001), 8563--8581.
\bibitem{KNY17}
Kajiwara K., Noumi M., and Yamada Y.,
{\it Geometric aspects of Painlev\'e equations}, J. Phys. A: Math. Theor. {\bf 50} (2017), 073001 (164pp) (Topical Review). %arXiv 1509.08186 [nlin.SI].
\bibitem{KS98}
Koekoek R., and Swarttouw R. F., {\it The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue}, Delft University of Technology, Department of Technical Mathematics and Informatics Report (1998) no. 98--17.
%\bibitem{KTGR00}
%Kruskal M. D., Tamizhmani K. M., Grammaticos B., and Ramani A., {\it Asymmetric discrete Painlev\'e equations}, Regul. Chaot. Dyn., {\bf 5} (2000), 273--281.
%\bibitem{LT14}
%Lisovyy Oleg., and Tykhyy Y., {\it 
%Algebraic solutions of the sixth Painlev\'e equation}. 
%J. Geom. Phys. {\bf 85} (2014), 124--163.
\bibitem{Magnus95}
Magnus A., {\it Painlev\'{e}-type differential equations for the recurrence coefficients of semi- classical orthogonal polynomials}, J. Comput. Appl. Math., {\bf 57} (1995), 215--237.
\bibitem{Mano12}
Mano T., {\it Determinant formula for solutions of the Garnier system and Pad\'e approximation}.
J. Phys. A: Math. Theor. {\bf 45} (2012), 135206--135219.
\bibitem{MT14}
Mano T., and Tsuda T.,
{\it  Two approximation problems by Hermite and the Schlesinger transformations} (Japanese),  
RIMS Kokyuroku Bessatsu {\bf B47} (2014), 77--86.
\bibitem{MT17}
Mano T., and Tsuda T., {\it Hermite-Pad\'e approximation, isomonodromic deformation and hypergeometric integral}. Math. Z. {\bf 285} (2017), no. 1--2, 397--431.
%\bibitem{Masuda03}
%Masuda T., {\it On a class of algebraic solutions to the Painlev\'e {\rm VI} equation, its determinant formula and coalescence cascade}, Funkcial. Ekvac. {\bf46} (2003), no.1, 121--171.
%\bibitem{MOK02}
%Masuda T., Ohta Y., and Kajiwara K., {\it A determinant formula for a class of rational solutions of Painlev\'e {\rm V} equation}. Nagoya Math. J. {\bf 168} (2002), 1--25. 
%\bibitem{Masuda09}
%Masuda T., {\it Hypergeometric $\tau$-functions of the q-Painlev\'e system of type $E^{(1)}_7$}, SIGMA, {\bf 5} (2009), 035 (30pp).
%\bibitem{Masuda11}
%Masuda T., {\it Hypergeometric $\tau$-functions of the q-Painlev\'e system of type $E^{(1)}_8$}, Ramanujan J. {\bf 24} (2011) 1--31.
\bibitem{MMT99}
Matano T., A. Matumiya A., and Takano K., {\it On some Hamiltonian structures of Painlev\'e systems, {\rm II}}. J. Math. Soc. Japan, {\bf 51} (1999) 843--866.
%\bibitem{MSY03}
%Murata M., Sakai H., and Yoneda J., {\it Riccati solutions of discrete Painlev\'e equations with Weyl group symmetry of type $E_8^{(1)}$}, J. Math. Phys. {\bf 44} (2003) 1396--1414.
%\bibitem{Murata85}
%Murata Y., {\it Rational solutions of the second and the fourth Painlev\'e equations}, 
%Funkcial. Ekvac., {\bf 28} (1985), pp. 1--32.
%\bibitem{Murata04}
%Murata M., {\it New expressions for discrete Painlev\'e equations}, Funkcial. Ekvac. {\bf 47} (2004) 291--305. 
%\bibitem{Murata09}
%Murata M., {\it Lax forms of the $q$-Painlev\'e equations}, J. Phys. A: Math. Theor., {\bf 42} (2009), 115201--115217.
\bibitem{Nagao15}
Nagao, H., {\it The Pad\'e interpolation method applied to $q$-Painlev\'e equations}, Lett. Math. Phys. {\bf 105} (2015), no. 4, 503--521.
\bibitem{Nagao16}
Nagao H., {\it  Lax pairs for additive difference Painlev\'e equations}, arXiv:1604.02530 [nlin.SI]
\bibitem{Nagao17-1}
Nagao H., {\it The Pad\'e interpolation method applied to $q$-Painlev\'e equations II (differential grid version)}, Lett. Math. Phys. {\bf 107} (2017), no. 1, 107--127.
\bibitem{Nagao17-2}
Nagao H., in preparation.
\bibitem{NY16}
Nagao H., and Yamada Y., {\it Study of $q$-Garnier system by Pad\'e method}, Funkcial. Ekvac. to appear arXiv:1601.01099 [nlin.SI].
\bibitem{NY17}
Nagao H., and Yamada Y., in preparation.
%\bibitem{Nakazono10}
%Nakazono N., {\it Hypergeometric $tau$ Functions of the $q$-Painlev\'e Systems of Type $(A_2+A_1)^{(1)}$}, SIGMA, {\bf 6} (2010), 084--099.
\bibitem{Nakazono13}
Nakazono N., {\it Solutions to discrete Painlev\'e systems arising from two types of orthogonal polynomials}, Reports of RIAM Symposium, {\bf 23}AO--S7 (2013), 35--41.
%\bibitem{Noumi04}
%Noumi M., {\it Painlev\'e Equations through Symmetry}, Translations of Mathematical Monographs, vol. {\bf 223}, American Mathematical Society, Providence, RI, 2004.
%\bibitem{Noumi07}
%Noumi M., {\it Special functions arising from discrete Painlev\'e equations: a survey.}, J. Comput. Appl. Math. {\bf 202} (2007), no. 1, 48--55.
\bibitem{NTY13}
Noumi M., Tsujimoto S., and Yamada Y., {\it Pad\'e interpolation for elliptic Painlev\'e equation}, Symmetries, integrable systems and representations, Springer Proc. Math. Stat., Volume {\bf 40} (2013), 463--482.
%\bibitem{NY98}
%Noumi M., and Yamada Y., {\it Affine Weyl groups, discrete dynamical systems and Painlev\'e equations}, Comm. Math. Phys., {\bf 199} (1998), pp. 281--295
%\bibitem{NY99}
%Noumi M., and Yamada Y., {\it Symmetries in the fourth Painlev\'e equation and Okamoto polynomials}. Nagoya Math. J. {\bf 153} (1999), 53--86.
%\bibitem{NY04}
%Noumi M., and Yamada Y., {\it Symmetries in Painlev\'e equations}, Sugaku Expositions {\bf 17} (2004) 203--218, Originally appeared in Japanese in Sugaku
%53 (2001) 62--75 (Translated by T. Masuda).
\bibitem{ORG01}
Ohta Y., Ramani A. and Grammaticos B., {\it An affine Weyl group approach to the eight-parameter discrete Painlev\'e equation}, J. Phys. {\bf A34} (2001) 10523--10532.
\bibitem{Okamoto79}
Okamoto K., {\it Sur les feuilletages associ\'es aux \'equations du second ordre \'a points critiques fix\'es de P. Painlev\'e}, Japan. J. Math. (N.S.) {\bf 5} (1979) 1--79.
%\bibitem{Okamoto87--1}
%Okamoto K., {\it Studies on the Painlev\'e equations. {\rm I}. Sixth Painlev\'e equation $P_{\rm VI}$}. Ann. Mat. Pura Appl. (4) {\bf 146} (1987), 337--381. 
%\bibitem{Okamoto87-2}
%Okamoto K., {\it Studies on the Painlev\'e equations. {\rm II}. Fifth Painlev\'e equation $P_{\rm V}$}. Japan. J. Math. (N.S.) {\bf 13} (1987), no. 1, 47--76.
%\bibitem{Okamoto86-3}
%Okamoto K., {\it Studies on the Painlev\'e equations. {\rm III}. Second and fourth Painlev\'e equation,  $P_{\rm II}$ and  $P_{\rm IV}$}. Math. Ann. {\bf 275} (1986), no. 2, 221--255.
%\bibitem{Okamoto87-4}
%Okamoto K., {\it Studies on the Painlev\'e equations. {\rm IV}. Third Painlev\'e equation $P_{\rm III}$}. Funkcial. Ekvac. {\bf 30} (1987), no. 2-3, 305--332.
%パンルヴェの論文はいつ？
\bibitem{OR16-1}
Ormerod, C.M., and Rains E.M., {\it Commutation Relations and Discrete Garnier Systems}, SIGMA {\bf 12} (2016), 110, 50 pages. 
\bibitem{OR16-2}
Ormerod, C.M., and Rains E.M., {\it An elliptic Garnier system}, arXiv:1607.07831 [nlin.SI]. 
\bibitem{OWF11}
Ormerod C.M., Witte N.S., and Forrester P.J., {\it Connection preserving deformations and q-semi-classical orthogonal polynomials}, Nonlinearity, {\bf 24} (2011), 2405--2434. 
%\bibitem{Painleve}
%Painlev\'e P., {\it Sur les \'{e}quations diff\'{e}retielles du second ordre dont l’int\'{e}grale g\'{e}n\'{e}rale et uniforme. Oeuvret. III}, pp. 187-271
%Painlev\'e P., “Memoire sur les \UTF{00E9}quations diff\UTF{00E9}rentielles dont l'int\UTF{00E9}grale g\UTF{00E9}n\UTF{00E9}rale est uniforme”, Bull. Soc. Math. Phys. France 28: (1900), 201--261
%Painlev\'e P., “Sur les \UTF{00E9}quations diff\UTF{00E9}rentielles du second ordre et d'ordre sup\UTF{00E9}rieur dont l'int\UTF{00E9}grale g\UTF{00E9}n\UTF{00E9}rale est uniforme”, Acta Math. 25: (1902), 1--85, 
%\bibitem{Rains05}
%Rains E., {\it Recurrences for elliptic hypergeometric integrals}, in: M. Noumi, K. Takasaki (Eds.), Elliptic Integrable Systems, Rokko Lectures in
%Mathematics, vol. {\bf 18}, Department Mathematics, Kobe University, 2005, pp. 183--199.
%\bibitem{RG96}
%Ramani A., and Grammaticos B., {\it Discrete Painlev\'e equations: coalescences, limits and degeneracies},
%Phys. A {\bf 228} (1996), no. 1-4, 160--171.
%\bibitem{RGH91}
%Ramani A., Grammaticos B., and Hietarinta J., {\it Discrete Versions of the Painlev\'e Equations} Phys. Rev.Lett. {\bf 67} (1991) 1829--32
%\bibitem{RGO01}
%Ramani A., Grammaticos B., and Ohta Y., {\it The q-Painlev\'e {\rm V} equation and its geometrical description},
%J. Phys. A:Math. Gen. {\bf 34} (2001), no. 11, 2505--2513.
%\bibitem{RGO01}
%Ramani A., Grammaticos B., and Ohta Y., {\it A Geometrical Description of the Discrete Painlev\'e VI and V Equations}, Commun. Math. Phys. {\bf 217} (2001) 315--329.
%\bibitem{RGP96}
%Ramani A., Grammaticos B., and Papageorgiou V., {\it Singularity confinement, Symmetries and Integrability of Difference Equations} (Est\'erel, PQ, 1994), CRM Proceedings of Lecture Notes, vol. {\bf 9}, American Mathematical Society, Providence, RI, 1996, pp. 303--318.
%\bibitem{RGS95}
%Ramani A., Grammaticos B., and Satsuma J., {\it 
%Bilinear discrete Painlev\'e equations}, 
%J. Phys. A, {\it 28} (1995), pp. 4655--4665
\bibitem{RGTT01}
Ramani A., Grammaticos B., Tamizhmani T., and Tamizhmani K.M., Special Function Solutions of the Discrete Painlev\'e Equations, Comput. Math. Appl., {\bf42} (2001), no. 3--5, 603--614.
%\bibitem{qRT}
%G.~R.~W.~quispel, J.~A.~G.~Roberts, C.~J.~Thompson,
%{\it Integrable mappings and soliton equations},
%Phys. Lett. {\bf A126} (1988) 419--421.
%\bibitem{Rui}
%S.~N.~M.~Ruijsenaars, 
%{\it First degree analytic difference equations and %integrable quantum systems},
%J. Math. Phys. {\bf 38} (1997) 1069--1146. 
%\bibitem{Rains}
%E.~Rains, 
%{\it Recurrences for elliptic hypergeometric integrals},
%Rokko Lectures in Mathematics {\bf 18} (2005) 183--199.
%\bibitem{Sakai98}
%Sakai H., {\it Casorati determinant solutions for the $q$-difference sixth Painlev\'e equations}, Nonlinearity, {\bf 11} (1998), 823--833.
\bibitem{Sakai01}
Sakai H., {\it Rational surfaces with affine root systems and geometry of the Painlev\'e equations}, Commun. Math. Phys., {\bf 220} (2001), 165--221.
\bibitem{Sakai05}
Sakai H., {\it A $q$-Analog of the Garnier System}, Funkcial. Ekvac. {\bf 48} (2005), 273--297.
%\bibitem{Sakai06}
%Sakai H., {\it Lax form of the $q$-Painlev\'e equation associated with the $A_2^{(1)}$ surface}, J. Phys. A: Math. Gen., {\bf 39} (2006), 12203--12210.
\bibitem{Sakai07}
Sakai H., {\it Problem: discrete Painlev\'e equations and their Lax forms}. (English summary) Algebraic, analytic and geometric aspects of complex differential equations and their deformations. Painlev\'e hierarchies, 195--208, RIMS Kokyuroku Bessatsu, {\bf B2}, Res. Inst. Math. Sci. (RIMS), Kyoto, 2007. 
\bibitem{ST97}
Shioda T., and Takano K., {\it On some Hamiltonian structures of Painlev\'e systems, {\rm I}.} Funkcial. Ekvac. {\bf 40} (1997) 271--291.
%\bibitem{Spi}
%V.~P.~Spiridonov,
%{\it Essays on the theory of elliptic hypergeometric functions}, 
%Uspekhi Matematicheskikh Nauk {\bf 63} (2008) 3--72.
%{\it Classical elliptic hypergeometric functions and their applications},
%Rokko Lectures in Mathematics {\bf 18} (2005) 253--287.
%\bibitem{SZ}
%V.~Spiridonov and A.~Zhedanov,
%{\it Spectral transformation chains and some new biorthogonal rational functions}, 
%Commun. Math. Phys. {\bf 210} (2000) 49--83.
%\bibitem{Suzuki15}
%Suzuki T., {\it A $q$-analogue of the Drinfeld-Sokolov hierarchy of type $A$ and $q$-Painlev\'e system}, AMS Contemp. Math. {\bf 651} (2015), 25--38.
%\bibitem{TTGR04}
%Tamizhmani K.M.; Tamizhmani T., Grammaticos B. and Ramani A., {\it Special solutions for discrete Painlev\'e equations. Discrete integrable systems}, 323--382, Lecture Notes in Phys., {\bf 644}, Springer, Berlin, 2004.
%\bibitem{Tsuda10}
%Tsuda T., {\it On an integrable system of q-difference equations satisfied by the universal characters: its Lax formalism and an application to $q$-Painlev\'e equations}, Comm. Math. Phys. {\bf 293} (2010), 347--359.
%\bibitem{Tsuji}
%S.~Tsujimoto,
%{\it Determinant solutions of the nonautonomous %discrete Toda equation associated with the %deautonomized discrete KP hierarchy},
%J. Syst. Sci. Complex. {\bf 23} (2010) 153--176.
%\bibitem{Umemura99}
%Umemura H., {\it Painlev\'e equations in the past 100 years}, Amer. Math. Soc. Transl. (2) 204 (2001) 81--110 (Originally appeared in Japanese in Sugaku {\bf 51} (1999) 395--420).
\bibitem{Van07}
Van W.A., {\it Discrete Painlev\'e equations for recurrence coefficients of orthogonal polynomials. Difference equations, special functions and orthogonal polynomials}, World Sci. Publ., Hackensack, NJ (2007), 687--725. 
\bibitem{Witte09}
Witte N.S., {\it Biorthogonal Systems on the Unit Circle, Regular Semiclassical Weights, and the Discrete Garnier Equations}, IMRN, {\bf 6}(2009), 988--1025. 
\bibitem{Witte15}
Witte N.S., {\it Semiclassical orthogonal polynomial systems on nonuniform lattices, deformations of the Askey table, and analogues of isomonodromy}, Nagoya Mathematical Journal, {\bf 219} (2015),127-234
\bibitem{WO12}
Witte N.S., and Ormerod C.M, {\it Construction of a Lax Pair for the $E_6^{(1)}$ $q$-Painlev\'e System}, SIGMA, {\bf 8} (2012), 097--123. 
\bibitem{Yamada09-1}
Yamada Y., {\it Pad\'e method to Painlev\'e equations}, Funkcial. Ekvac., {\bf 52} (2009), 83--92.
\bibitem{Yamada09-2} 
Yamada Y., {\it A Lax formalism for the elliptic difference Painlev\'e equation}, SIGMA {\bf 5} (2009) 042 (15pp).
\bibitem{Yamada11}
Yamada Y., {\it Lax formalism for $q$-Painlev\'e equations with affine Weyl group symmetry of type $E^{(1)}_n$}, IMRN, {\bf 17} (2011), 3823--3838.
\bibitem{Yamada14}
Yamada Y., {\it A simple expression for discrete Painlev\'e equations}, RIMS Kokyuroku Bessatsu, B{\bf47} (2014), 087--095.
%\bibitem{Yamasaki09}
%Yamasaki K., {\it Pad\'e approximation and special solution for $q$-Painlev\'e VI equation}. Master thesis in Kobe University (Japanese) (2009)
%\bibitem{Yoshioka10}
%Yoshioka R., {\it A construction of special solution for $q$-Painlev\'e VI equation by the Pad\'e approximation}. Master thesis in Kobe University (Japanese) (2010)
%\bibitem{Zhedanov05}
%Zhedanov A.S., {\it Pad\'e interpolation table and biorthogonal rational functions}, Rokko Lectures in Mathematics, {\bf 18} (2005), 323--363.

\end{thebibliography}

\end{document}



1706.10101.extracted_bib
{Andrews72 %Andrews G.E., {Summation and transformation for basic Appell series, J. London Math. Soc. {4 (2) (1972), 618--622. %
1706.10101.extracted_bib
{Andrews75 %Andrews G.E., {Problems and prospectives for basic hypergeometric functions, in: Theory and application of special functions, Proc. Advanced Sem. Math. Res. Center, University of Wisconsin, Madison, WI, (1975), 19--224.
1706.10101.extracted_bib
{AB06 Arinkin D., and Borodin A., {Moduli spaces of d-connections and difference Painlev\'e equations Duke Mathematics Journal {134, no. 2 (2006): 515--56.
1706.10101.extracted_bib
{Bailey35 Bailey W., N., {Generalized hypergeometric series, Cambridge tracts in mathematics and mathematical physics No. 32, Cambridge, Cambridge University Press, (1935). %
1706.10101.extracted_bib
{Boalch06 %Boalch P., {The fifty-two icosahedral solutions to Painlev\'e {\rm VI. J. Reine Angew. Math. {596 (2006), 183--214.
1706.10101.extracted_bib
{Boalch09 Boalch P., {Quivers and difference Painlev\'e equations Groups and Symmetries 25--51, CRM Proceedings in Lecture Notes {47, Providence, RI: American Mathematical Society, (2009).
1706.10101.extracted_bib
{Clarkson13 Clarkson P.A., {Recurrence coefficients for discrete orthonormal polynomials and the Painlev\'e equations, J. Phys., A {46, no.18, (2013), 185205--185222.
1706.10101.extracted_bib
{Conte99 Conte R. (Ed.), {The Painlev\'e Property--One Century Later, CRM Series in Mathematical Physics, Springer, Berlin, 1999. %
1706.10101.extracted_bib
{DM00 %Dubrovin B., and Mazzocco M., {Monodromy of certain Painlev\'e {\rm VI transcendents and reflection groups, Invent. Math. {141 (2000) 55--147.
1706.10101.extracted_bib
{DST13 Dzhamay A., Sakai H., and Takenawa T., {Discrete Hamiltonian Structure of Schlesinger Transformations, arXiv:1302.2972 [math-ph].
1706.10101.extracted_bib
{DT14 Dzhamay A., and Takenawa T., {Geometric Analysis of Reductions from Schlesinger transformations to difference Painlev\'e equations, arXiv:1408.3778 [math-ph].
1706.10101.extracted_bib
{Forrester10 Forrester P. J., {Log-Gases and Random Matrices, London Mathematical Society Monographs {34 (Princeton: Princeton University Press, 2010).
1706.10101.extracted_bib
{Fuchs1907 Fuchs R., \``{Uber lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegene wesentlich singul\``{aren Stellen. Math. Ann. 63, (1907) 301--321 %
1706.10101.extracted_bib
{Gambier1910 %Gambier B., {Sur les \'{equations diff\'{eretielles du second ordre et du premier degr\'{e dont l’int\'{egrale g\'{en\'{erale est \'{a points critiques fixes. Acta. Math. {33, (1910) 1--55 %
1706.10101.extracted_bib
{Garnier12 %Garnier R., {Sur des \'{equations diff\'{erentielles du troisi\'{eme ordre dont l’int\'{egrale g\'{en\'{erale est uniforme et sur une classe d’\'{equations nouvelles d’ordre sup\'{erieur dont l’int\'{egrale g\'{en\'{erale a ses points critiques fixes. Ann. Sci. Ecole Norm. Super. {29, (1912) 1--126 %
1706.10101.extracted_bib
{GaR90 %Gasper G., and Rahman M., {Basic Hypergeometric Series, Encyclopedia of Mathematics %and Its Applications, Volume {35 Cambridge University Press, Cambridge, (1990).
1706.10101.extracted_bib
{GaR04 Gasper G., and Rahman M., {Basic Hypergeometric Series. With a foreword by Richard Askey. Second edition. Encyclopedia of Mathematics and its Applications, Volume {96, Cambridge University Press, Cambridge, (2004).
1706.10101.extracted_bib
{GORS98 Grammaticos B., Ohta Y., Ramani A., and Sakai H., {Degeneration through coalescence of the $q$-Painlev\'e {\rm VI equations J. Phys. A : Math. Gen. {31(15), 3545--3558, (1998) %
1706.10101.extracted_bib
{GNPRS94 %Grammaticos B., Nijhoff F.W., Papageorgiou V.G., Ramani A., and Satsuma J., {Linearization and solution of the discrete Painlev\'e III equation, Phys. Lett. A {185, (1994), 446. %
1706.10101.extracted_bib
{GNR99 %Grammaticos B., Nijhoff F.W., and Ramani A., {Discrete Painlev\'e equations, The Painlev\'e Property: One Century Later, Editor R. Conte, CRM Ser. Math. Phys., Springer, New York, (1999), 413--516.
1706.10101.extracted_bib
{GR99 Grammaticos B., and Ramani A., {On a novel $q$-discrete analogue of the Painlev\'{e \rm{VI equation, Phys. Lett., A{257 (1999), 288--292. %
1706.10101.extracted_bib
{GR04 %Grammaticos B., and A. Ramani, {Discrete Painlev\'e equations: a review, in: Grammaticos B., Kosmann-Schwarzbach Y., Tamizhmani T. (Eds.), Discrete Integrable Systems, Lecture Notes in Physics, vol. {644, Springer, Berlin, 2004, pp. 245--321. %
1706.10101.extracted_bib
{GRO03 %Grammaticos B., Ramani A., Ohta Y., {A Unified Description of the Asymmetric $q$-$P_{V$ and $d$-$P_{IV$ Equations and their Schlesinger Transformations. J. Nonlinear Math. Phys. {10(2), 215--228 (2003) %
1706.10101.extracted_bib
{GRP91 %Grammaticos B., Ramani A., and Papageorgiou V. G., {Do Integrable Mappings Have the Painlev\'e Property? Phys. Rev. Lett. {67 1825--18258 (1991) %
1706.10101.extracted_bib
{GL82 %Gromak V.I. and Lukashevich N.A., {Special class of solutions of Painlev\'e equations, Differential Equations {18 (1982) 317--326. %
1706.10101.extracted_bib
{GLS02 %Gromak V.I., Laine I. and Shimomura S., {Painlev\'e Differential Equations in the Complex Plane, de Gruyter Studies in Mathematics, vol. {28, Walter de Gruyter \& Co., Berlin, 2002. %
1706.10101.extracted_bib
{HK07 %Hamamoto T., and Kajiwara K., {Hypergeometric solutions to the q-Painlev\'e equation of type $A_4^{(1)$, J. Phys. A: Math. Theor., {40 (2007), 12509--12524.
1706.10101.extracted_bib
{Ikawa13 Ikawa Y., {Hypergeometric Solutions for the $q$-Painlev\'e Equation of Type $E_6^{(1)$ by the Pad\'e method, Lett. Math. Phys., Volume {103, Issue 7 (2013), 743--763.
1706.10101.extracted_bib
{IKSY91 Iwasaki K., Kimura H., Shimomura S. and Yoshida M., {From Gauss to Painlev\'e--A Modern Theory of Special Functions, Aspects of Mathematics, vol. {E16, Vieweg, 1991.
1706.10101.extracted_bib
{Jacobi Jacobi,C.G.J., {\``{Uber die Darstellung einer Reihe gegebner Werthe durch eine gebrochne rationale Function, J. Reine Angew. Math., {30 (1846) 127--156. %
1706.10101.extracted_bib
{JM81-1 %Jimbo M., Miwa T., and Ueno K., {Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. Physica {2D (1981) 306--52
1706.10101.extracted_bib
{JM81-2 Jimbo M., and Miwa T., {Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II. Physica {2D (1981) 407-448 %
1706.10101.extracted_bib
{JM81-3 %Jimbo M., and Miwa T., {Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. III.Physica {4D (1981) 26--46 %
1706.10101.extracted_bib
{JS96 %Jimbo M., and Sakai H., {A $q$-analog of the sixth Painlev\'e equation, Lett. Math. Phys., {38 (1996), 145--154. %
1706.10101.extracted_bib
{Kajihara04 %Kajihara Y., {Euler transformation formula for multiple basic hypergeometric series of type $A$ and some applications, Advances in Mathematics {187 Issue 1, (2004), 53--97.
1706.10101.extracted_bib
{Kajiwara08 Kajiwara K., {Hypergeometric solutions to the additive discrete Painlev\'e equations with affine Weyl group symmetry of type E, Reports of Research Institute for Applied Mechanics Symposium, No.19ME{S2 (2008), No.3. %
1706.10101.extracted_bib
{KK03 %Kajiwara K., and Kimura K., {On a q-difference Painlev\'e III equation, I.Derivation, symmetry and Riccati type solutions. J. Nonlinear Math. Phys.{10 (2003) 86--102. %
1706.10101.extracted_bib
{Kitaev06 %Kitaev A.V., {Remarks towards a classification of $RS^2_4(3)$-transformations and algebraic solutions of the sixth Painlev\'e equation. Th\'eories asymptotiques et \'equations de Painlev\'e, 199--227, S\'emin. Congr., {14, Soc. Math. France, Paris, 2006. %
1706.10101.extracted_bib
{KLM94 %Kitaev A.V., Law C.K., and McLeod J.B., %{Rational solutions of the fifth Painlev\'e equation, Differential Integral Equations, {7 (1994), pp. 967--1000 %
1706.10101.extracted_bib
{KMNOY03 %Kajiwara K., Masuda T., Noumi M., Ohta Y., and Yamada Y., ${_{10E_{9$ solution to the elliptic Painlev\'e equation, J. Phys. A: Math. Gen. {36 (2003), L263-L272, nlin.SI/0303032. %
1706.10101.extracted_bib
{KMNOY04 %Kajiwara K., Masuda T., Noumi M., Ohta Y., and Yamada Y., %{Hypergeometric solutions to the $q$-Painlev\'e equations, %Int. Math. Res. Not. 2004, {47 (2004), 2497--2521. %
1706.10101.extracted_bib
{KMNOY05 %Kajiwara K., Masuda T., Noumi M., Ohta Y., and Yamada Y., {Construction of hypergeometric solutions to the $q$-Painlev\'e equations, Int. Math. Res. Not. (2005) 1439--1463 %
1706.10101.extracted_bib
{KMNOY06 %Kajiwara K., Masuda T., Noumi M., Ohta Y., and Yamada Y., {Point configurations, Cremona transformations and the elliptic difference Painlev\'e equation. (English, French summary) Th\'eories asymptotiques et \'equations de Painlev\'e, 169--198, %S\'emin. Congr., {14, Soc. Math. France, Paris, 2006. %
1706.10101.extracted_bib
{KNY01 %Kajiwara K., Noumi M., and Yamada Y., {A study on the fourth $q$-Painlev\'e equation, J. Phys. A: Math. Gen., {34 (2001), 8563--8581.
1706.10101.extracted_bib
{KNY17 Kajiwara K., Noumi M., and Yamada Y., {Geometric aspects of Painlev\'e equations, J. Phys. A: Math. Theor. {50 (2017), 073001 (164pp) (Topical Review). %arXiv 1509.08186 [nlin.SI].
1706.10101.extracted_bib
{KS98 Koekoek R., and Swarttouw R. F., {The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Delft University of Technology, Department of Technical Mathematics and Informatics Report (1998) no. 98--17. %
1706.10101.extracted_bib
{KTGR00 %Kruskal M. D., Tamizhmani K. M., Grammaticos B., and Ramani A., {Asymmetric discrete Painlev\'e equations, Regul. Chaot. Dyn., {5 (2000), 273--281. %
1706.10101.extracted_bib
{LT14 %Lisovyy Oleg., and Tykhyy Y., {%Algebraic solutions of the sixth Painlev\'e equation. %J. Geom. Phys. {85 (2014), 124--163.
1706.10101.extracted_bib
{Magnus95 Magnus A., {Painlev\'{e-type differential equations for the recurrence coefficients of semi- classical orthogonal polynomials, J. Comput. Appl. Math., {57 (1995), 215--237.
1706.10101.extracted_bib
{Mano12 Mano T., {Determinant formula for solutions of the Garnier system and Pad\'e approximation. J. Phys. A: Math. Theor. {45 (2012), 135206--135219.
1706.10101.extracted_bib
{MT14 Mano T., and Tsuda T., {Two approximation problems by Hermite and the Schlesinger transformations (Japanese), RIMS Kokyuroku Bessatsu {B47 (2014), 77--86.
1706.10101.extracted_bib
{MT17 Mano T., and Tsuda T., {Hermite-Pad\'e approximation, isomonodromic deformation and hypergeometric integral. Math. Z. {285 (2017), no. 1--2, 397--431. %
1706.10101.extracted_bib
{Masuda03 %Masuda T., {On a class of algebraic solutions to the Painlev\'e {\rm VI equation, its determinant formula and coalescence cascade, Funkcial. Ekvac. {46 (2003), no.1, 121--171. %
1706.10101.extracted_bib
{MOK02 %Masuda T., Ohta Y., and Kajiwara K., {A determinant formula for a class of rational solutions of Painlev\'e {\rm V equation. Nagoya Math. J. {168 (2002), 1--25. %
1706.10101.extracted_bib
{Masuda09 %Masuda T., {Hypergeometric $\tau$-functions of the q-Painlev\'e system of type $E^{(1)_7$, SIGMA, {5 (2009), 035 (30pp). %
1706.10101.extracted_bib
{Masuda11 %Masuda T., {Hypergeometric $\tau$-functions of the q-Painlev\'e system of type $E^{(1)_8$, Ramanujan J. {24 (2011) 1--31.
1706.10101.extracted_bib
{MMT99 Matano T., A. Matumiya A., and Takano K., {On some Hamiltonian structures of Painlev\'e systems, {\rm II. J. Math. Soc. Japan, {51 (1999) 843--866. %
1706.10101.extracted_bib
{MSY03 %Murata M., Sakai H., and Yoneda J., {Riccati solutions of discrete Painlev\'e equations with Weyl group symmetry of type $E_8^{(1)$, J. Math. Phys. {44 (2003) 1396--1414. %
1706.10101.extracted_bib
{Murata85 %Murata Y., {Rational solutions of the second and the fourth Painlev\'e equations, %Funkcial. Ekvac., {28 (1985), pp. 1--32. %
1706.10101.extracted_bib
{Murata04 %Murata M., {New expressions for discrete Painlev\'e equations, Funkcial. Ekvac. {47 (2004) 291--305. %
1706.10101.extracted_bib
{Murata09 %Murata M., {Lax forms of the $q$-Painlev\'e equations, J. Phys. A: Math. Theor., {42 (2009), 115201--115217.
1706.10101.extracted_bib
{Nagao15 Nagao, H., {The Pad\'e interpolation method applied to $q$-Painlev\'e equations, Lett. Math. Phys. {105 (2015), no. 4, 503--521.
1706.10101.extracted_bib
{Nagao16 Nagao H., {Lax pairs for additive difference Painlev\'e equations, arXiv:1604.02530 [nlin.SI]
1706.10101.extracted_bib
{Nagao17-1 Nagao H., {The Pad\'e interpolation method applied to $q$-Painlev\'e equations II (differential grid version), Lett. Math. Phys. {107 (2017), no. 1, 107--127.
1706.10101.extracted_bib
{Nagao17-2 Nagao H., in preparation.
1706.10101.extracted_bib
{NY16 Nagao H., and Yamada Y., {Study of $q$-Garnier system by Pad\'e method, Funkcial. Ekvac. to appear arXiv:1601.01099 [nlin.SI].
1706.10101.extracted_bib
{NY17 Nagao H., and Yamada Y., in preparation. %
1706.10101.extracted_bib
{Nakazono10 %Nakazono N., {Hypergeometric $tau$ Functions of the $q$-Painlev\'e Systems of Type $(A_2+A_1)^{(1)$, SIGMA, {6 (2010), 084--099.
1706.10101.extracted_bib
{Nakazono13 Nakazono N., {Solutions to discrete Painlev\'e systems arising from two types of orthogonal polynomials, Reports of RIAM Symposium, {23AO--S7 (2013), 35--41. %
1706.10101.extracted_bib
{Noumi04 %Noumi M., {Painlev\'e Equations through Symmetry, Translations of Mathematical Monographs, vol. {223, American Mathematical Society, Providence, RI, 2004. %
1706.10101.extracted_bib
{Noumi07 %Noumi M., {Special functions arising from discrete Painlev\'e equations: a survey., J. Comput. Appl. Math. {202 (2007), no. 1, 48--55.
1706.10101.extracted_bib
{NTY13 Noumi M., Tsujimoto S., and Yamada Y., {Pad\'e interpolation for elliptic Painlev\'e equation, Symmetries, integrable systems and representations, Springer Proc. Math. Stat., Volume {40 (2013), 463--482. %
1706.10101.extracted_bib
{NY98 %Noumi M., and Yamada Y., {Affine Weyl groups, discrete dynamical systems and Painlev\'e equations, Comm. Math. Phys., {199 (1998), pp. 281--295 %
1706.10101.extracted_bib
{NY99 %Noumi M., and Yamada Y., {Symmetries in the fourth Painlev\'e equation and Okamoto polynomials. Nagoya Math. J. {153 (1999), 53--86. %
1706.10101.extracted_bib
{NY04 %Noumi M., and Yamada Y., {Symmetries in Painlev\'e equations, Sugaku Expositions {17 (2004) 203--218, Originally appeared in Japanese in Sugaku %53 (2001) 62--75 (Translated by T. Masuda).
1706.10101.extracted_bib
{ORG01 Ohta Y., Ramani A. and Grammaticos B., {An affine Weyl group approach to the eight-parameter discrete Painlev\'e equation, J. Phys. {A34 (2001) 10523--10532.
1706.10101.extracted_bib
{Okamoto79 Okamoto K., {Sur les feuilletages associ\'es aux \'equations du second ordre \'a points critiques fix\'es de P. Painlev\'e, Japan. J. Math. (N.S.) {5 (1979) 1--79. %
1706.10101.extracted_bib
{Okamoto87--1 %Okamoto K., {Studies on the Painlev\'e equations. {\rm I. Sixth Painlev\'e equation $P_{\rm VI$. Ann. Mat. Pura Appl. (4) {146 (1987), 337--381. %
1706.10101.extracted_bib
{Okamoto87-2 %Okamoto K., {Studies on the Painlev\'e equations. {\rm II. Fifth Painlev\'e equation $P_{\rm V$. Japan. J. Math. (N.S.) {13 (1987), no. 1, 47--76. %
1706.10101.extracted_bib
{Okamoto86-3 %Okamoto K., {Studies on the Painlev\'e equations. {\rm III. Second and fourth Painlev\'e equation, $P_{\rm II$ and $P_{\rm IV$. Math. Ann. {275 (1986), no. 2, 221--255. %
1706.10101.extracted_bib
{Okamoto87-4 %Okamoto K., {Studies on the Painlev\'e equations. {\rm IV. Third Painlev\'e equation $P_{\rm III$. Funkcial. Ekvac. {30 (1987), no. 2-3, 305--332. %パンルヴェの論文はいつ？
1706.10101.extracted_bib
{OR16-1 Ormerod, C.M., and Rains E.M., {Commutation Relations and Discrete Garnier Systems, SIGMA {12 (2016), 110, 50 pages.
1706.10101.extracted_bib
{OR16-2 Ormerod, C.M., and Rains E.M., {An elliptic Garnier system, arXiv:1607.07831 [nlin.SI].
1706.10101.extracted_bib
{OWF11 Ormerod C.M., Witte N.S., and Forrester P.J., {Connection preserving deformations and q-semi-classical orthogonal polynomials, Nonlinearity, {24 (2011), 2405--2434. %
1706.10101.extracted_bib
{Painleve %Painlev\'e P., {Sur les \'{equations diff\'{eretielles du second ordre dont l’int\'{egrale g\'{en\'{erale et uniforme. Oeuvret. III, pp. 187-271 %Painlev\'e P., “Memoire sur les \UTF{00E9quations diff\UTF{00E9rentielles dont l'int\UTF{00E9grale g\UTF{00E9n\UTF{00E9rale est uniforme”, Bull. Soc. Math. Phys. France 28: (1900), 201--261 %Painlev\'e P., “Sur les \UTF{00E9quations diff\UTF{00E9rentielles du second ordre et d'ordre sup\UTF{00E9rieur dont l'int\UTF{00E9grale g\UTF{00E9n\UTF{00E9rale est uniforme”, Acta Math. 25: (1902), 1--85, %
1706.10101.extracted_bib
{Rains05 %Rains E., {Recurrences for elliptic hypergeometric integrals, in: M. Noumi, K. Takasaki (Eds.), Elliptic Integrable Systems, Rokko Lectures in %Mathematics, vol. {18, Department Mathematics, Kobe University, 2005, pp. 183--199. %
1706.10101.extracted_bib
{RG96 %Ramani A., and Grammaticos B., {Discrete Painlev\'e equations: coalescences, limits and degeneracies, %Phys. A {228 (1996), no. 1-4, 160--171. %
1706.10101.extracted_bib
{RGH91 %Ramani A., Grammaticos B., and Hietarinta J., {Discrete Versions of the Painlev\'e Equations Phys. Rev.Lett. {67 (1991) 1829--32 %
1706.10101.extracted_bib
{RGO01 %Ramani A., Grammaticos B., and Ohta Y., {The q-Painlev\'e {\rm V equation and its geometrical description, %J. Phys. A:Math. Gen. {34 (2001), no. 11, 2505--2513. %
1706.10101.extracted_bib
{RGO01 %Ramani A., Grammaticos B., and Ohta Y., {A Geometrical Description of the Discrete Painlev\'e VI and V Equations, Commun. Math. Phys. {217 (2001) 315--329. %
1706.10101.extracted_bib
{RGP96 %Ramani A., Grammaticos B., and Papageorgiou V., {Singularity confinement, Symmetries and Integrability of Difference Equations (Est\'erel, PQ, 1994), CRM Proceedings of Lecture Notes, vol. {9, American Mathematical Society, Providence, RI, 1996, pp. 303--318. %
1706.10101.extracted_bib
{RGS95 %Ramani A., Grammaticos B., and Satsuma J., {%Bilinear discrete Painlev\'e equations, %J. Phys. A, {28 (1995), pp. 4655--4665
1706.10101.extracted_bib
{RGTT01 Ramani A., Grammaticos B., Tamizhmani T., and Tamizhmani K.M., Special Function Solutions of the Discrete Painlev\'e Equations, Comput. Math. Appl., {42 (2001), no. 3--5, 603--614. %
1706.10101.extracted_bib
{qRT %G.~R.~W.~quispel, J.~A.~G.~Roberts, C.~J.~Thompson, %{Integrable mappings and soliton equations, %Phys. Lett. {A126 (1988) 419--421. %
1706.10101.extracted_bib
{Rui %S.~N.~M.~Ruijsenaars, %{First degree analytic difference equations and %integrable quantum systems, %J. Math. Phys. {38 (1997) 1069--1146. %
1706.10101.extracted_bib
{Rains %E.~Rains, %{Recurrences for elliptic hypergeometric integrals, %Rokko Lectures in Mathematics {18 (2005) 183--199. %
1706.10101.extracted_bib
{Sakai98 %Sakai H., {Casorati determinant solutions for the $q$-difference sixth Painlev\'e equations, Nonlinearity, {11 (1998), 823--833.
1706.10101.extracted_bib
{Sakai01 Sakai H., {Rational surfaces with affine root systems and geometry of the Painlev\'e equations, Commun. Math. Phys., {220 (2001), 165--221.
1706.10101.extracted_bib
{Sakai05 Sakai H., {A $q$-Analog of the Garnier System, Funkcial. Ekvac. {48 (2005), 273--297. %
1706.10101.extracted_bib
{Sakai06 %Sakai H., {Lax form of the $q$-Painlev\'e equation associated with the $A_2^{(1)$ surface, J. Phys. A: Math. Gen., {39 (2006), 12203--12210.
1706.10101.extracted_bib
{Sakai07 Sakai H., {Problem: discrete Painlev\'e equations and their Lax forms. (English summary) Algebraic, analytic and geometric aspects of complex differential equations and their deformations. Painlev\'e hierarchies, 195--208, RIMS Kokyuroku Bessatsu, {B2, Res. Inst. Math. Sci. (RIMS), Kyoto, 2007.
1706.10101.extracted_bib
{ST97 Shioda T., and Takano K., {On some Hamiltonian structures of Painlev\'e systems, {\rm I. Funkcial. Ekvac. {40 (1997) 271--291. %
1706.10101.extracted_bib
{Spi %V.~P.~Spiridonov, %{Essays on the theory of elliptic hypergeometric functions, %Uspekhi Matematicheskikh Nauk {63 (2008) 3--72. %{Classical elliptic hypergeometric functions and their applications, %Rokko Lectures in Mathematics {18 (2005) 253--287. %
1706.10101.extracted_bib
{SZ %V.~Spiridonov and A.~Zhedanov, %{Spectral transformation chains and some new biorthogonal rational functions, %Commun. Math. Phys. {210 (2000) 49--83. %
1706.10101.extracted_bib
{Suzuki15 %Suzuki T., {A $q$-analogue of the Drinfeld-Sokolov hierarchy of type $A$ and $q$-Painlev\'e system, AMS Contemp. Math. {651 (2015), 25--38. %
1706.10101.extracted_bib
{TTGR04 %Tamizhmani K.M.; Tamizhmani T., Grammaticos B. and Ramani A., {Special solutions for discrete Painlev\'e equations. Discrete integrable systems, 323--382, Lecture Notes in Phys., {644, Springer, Berlin, 2004. %
1706.10101.extracted_bib
{Tsuda10 %Tsuda T., {On an integrable system of q-difference equations satisfied by the universal characters: its Lax formalism and an application to $q$-Painlev\'e equations, Comm. Math. Phys. {293 (2010), 347--359. %
1706.10101.extracted_bib
{Tsuji %S.~Tsujimoto, %{Determinant solutions of the nonautonomous %discrete Toda equation associated with the %deautonomized discrete KP hierarchy, %J. Syst. Sci. Complex. {23 (2010) 153--176. %
1706.10101.extracted_bib
{Umemura99 %Umemura H., {Painlev\'e equations in the past 100 years, Amer. Math. Soc. Transl. (2) 204 (2001) 81--110 (Originally appeared in Japanese in Sugaku {51 (1999) 395--420).
1706.10101.extracted_bib
{Van07 Van W.A., {Discrete Painlev\'e equations for recurrence coefficients of orthogonal polynomials. Difference equations, special functions and orthogonal polynomials, World Sci. Publ., Hackensack, NJ (2007), 687--725.
1706.10101.extracted_bib
{Witte09 Witte N.S., {Biorthogonal Systems on the Unit Circle, Regular Semiclassical Weights, and the Discrete Garnier Equations, IMRN, {6(2009), 988--1025.
1706.10101.extracted_bib
{Witte15 Witte N.S., {Semiclassical orthogonal polynomial systems on nonuniform lattices, deformations of the Askey table, and analogues of isomonodromy, Nagoya Mathematical Journal, {219 (2015),127-234
1706.10101.extracted_bib
{WO12 Witte N.S., and Ormerod C.M, {Construction of a Lax Pair for the $E_6^{(1)$ $q$-Painlev\'e System, SIGMA, {8 (2012), 097--123.
1706.10101.extracted_bib
{Yamada09-1 Yamada Y., {Pad\'e method to Painlev\'e equations, Funkcial. Ekvac., {52 (2009), 83--92.
1706.10101.extracted_bib
{Yamada09-2 Yamada Y., {A Lax formalism for the elliptic difference Painlev\'e equation, SIGMA {5 (2009) 042 (15pp).
1706.10101.extracted_bib
{Yamada11 Yamada Y., {Lax formalism for $q$-Painlev\'e equations with affine Weyl group symmetry of type $E^{(1)_n$, IMRN, {17 (2011), 3823--3838.
1706.10101.extracted_bib
{Yamada14 Yamada Y., {A simple expression for discrete Painlev\'e equations, RIMS Kokyuroku Bessatsu, B{47 (2014), 087--095. %
1706.10101.extracted_bib
{Yamasaki09 %Yamasaki K., {Pad\'e approximation and special solution for $q$-Painlev\'e VI equation. Master thesis in Kobe University (Japanese) (2009) %
1706.10101.extracted_bib
{Yoshioka10 %Yoshioka R., {A construction of special solution for $q$-Painlev\'e VI equation by the Pad\'e approximation. Master thesis in Kobe University (Japanese) (2010) %
1706.10101.extracted_bib
{Zhedanov05 %Zhedanov A.S., {Pad\'e interpolation table and biorthogonal rational functions, Rokko Lectures in Mathematics, {18 (2005), 323--363.