Information on some recent applications of umbral extensions to discrete mathematics
A.K.Kwa\'sniewski
%%% ENDNAME
Higher School of Mathematics and Applied Informatics
PL - 15-021 Bialystok
ul.Kamienna 17
Poland
At the first part of our communicate we show how specific umbral extensions of the Stirling numbers of the second kind result in new type of Dobinski-like formulas. In the second part among others one recovers how and why Morgan Ward solution of uncountable family of $\psi$- {\em difference calculus} nonhomogeneous equations $\Delta_{\psi}f=\varphi$ in the form $$f(x)=\sum_{n \geq 1}\frac{B_{n}}{n_{\psi}!}\varphi^{(n-1)}(x)+\int_{\psi}\varphi(x) +p(x)$$ extends to $\psi$- Appell polynomials case automatically. Illustrative specifications to $q$-calculus case and Fibonomial calculus case are made explicit due to the usage of upside down notation for objects of Extended Finite Operator Calculus .
math0411145.tex



\documentclass[11pt]{article}


\title{ Information on some recent applications of  umbral extensions to discrete mathematics}
\author{A.K.Kwa\'sniewski\\  %%% ENDNAME
\\ Higher School of Mathematics and Applied Informatics\\
PL - 15-021 Bialystok , ul.Kamienna 17,  Poland
\\e-mail: kwandr@wp.pl}


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\begin{document}
\maketitle\ \textit{to appear in  Review Bulletin of Calcutta
Mathematical Society Vol. 13 (2005)}
\begin{abstract}
At the first part of our communicate we show how specific umbral
extensions of the Stirling numbers of the second kind  result in
new type  of Dobinski-like formulas. In the second part among
others one recovers how and why  Morgan Ward solution of
uncountable family of  $\psi$- {\em difference calculus}
nonhomogeneous equations $\Delta_{\psi}f=\varphi$ in the form
$$f(x)=\sum_{n \geq
1}\frac{B_{n}}{n_{\psi}!}\varphi^{(n-1)}(x)+\int_{\psi}\varphi(x)
+p(x)$$ extends to $\psi$- Appell polynomials case automatically.
Illustrative specifications to $q$-calculus case and Fibonomial
calculus case are made explicit due to the usage of upside down
notation for objects of Extended Finite Operator Calculus .

\end{abstract}

MCS numbers:  05A40, 11B73, 81S99

Key words: extended umbral calculus,Dobinski type
formulas,difference equations

\vspace{0.2cm} to be presented at the ISRAMA Congress, Calcuta -
India , December 2004


\vspace{0.5cm}
\subsection*{INTRODUCTION}

The upside down convenient notation for objects of Extended Finite
Operator Calculus  \textbf{(EFOC) } recently being developed and
promoted by the present author \cite{1,2,3,4} is to be used
throughout the whole exposition of both parts.\\

\vspace{2mm}

The \textbf{EFOC } is an implementation of operator approach of
Gian Carlo Rota to various by now formulations of extended of
umbral difference calculi. For abundant references - thousands  of
them - see \cite{5} and \cite{1,2,3,4}.

\vspace{2mm}

In \cite{1,2,3,4,5} and here also $\psi$ denotes  a number or
functions` sequence - sequence of functions of a parameter $q$.
$\psi$ constitutes the Morgan Ward [6] extension (see: after then
Viskov [7])  of $\langle\frac{1}{n!}\rangle_{n\geq 0}$ sequence to
quite arbitrary one (the so called - "admissible" in [8] and after
then in [1,2,3,4,5]).The specific choices are for example :
Fibonomialy-extended sequence $\langle\frac{1}{F_n!}\rangle_{n\geq
0}$ ($\langle F_n \rangle$ - Fibonacci sequence )  or just "the
usual" $\psi$-sequence $\langle\frac{1}{n!}\rangle_{n\geq 0}$ or
the famous Gauss $q$-extended $\langle\frac{1}{n_q!}\rangle_{n\geq
0}$ admissible sequence of extended umbral operator calculus,
where $ n_q=\frac{1-q^n}{1-q}$ and $n_q!=n_q(n-1)_q! , 0_q!=1$ -
see more below. With such type extension we frequently may in a
"$\psi$-mnemonic" way repeat reasoning and this what was done by
Rota (see at first \textbf{(FOC) } of Rota in [9]).

\vspace{2mm}

The simplicity of the first steps to be done while identifying
general properties of  $\psi$-extended objects  consists in
writing objects of these extensions in mnemonic convenient
\textbf{upside down notation} \cite{1,2,3,4} , \cite{10} which is
here down introduced as follows:

\begin{equation}\label{eq1}
\frac {\psi_{(n-1)}}{\psi_n}\equiv n_\psi,
n_\psi!=n_\psi(n-1)_\psi!, n>0 ,   x_{\psi}\equiv \frac
{\psi{(x-1)}}{\psi(x)} ,
\end{equation}


\begin{equation}\label{eq2}
x_{\psi}^{\underline{k}}=x_{\psi}(x-1)_\psi(x-2)_{\psi}...(x-k+1)_{\psi}
\end{equation}


\begin{equation}\label{eq3}
x_{\psi}(x-1)_{\psi}...(x-k+1)_{\psi}=
\frac{\psi(x-1)\psi(x-2)...\psi(x-k)} {\psi(x)
\psi(x-1)...\psi(x-k +1)} .
\end{equation}


If one writes the above in the form $x_{\psi} \equiv \frac
{\psi{(x-1)}}{\psi(x)}\equiv \Phi(x)\equiv\Phi_x\equiv x_{\Phi}$ ,
one sees that the name upside down notation is legitimate.

 You may consult \cite{1,2,3,4,5} and \cite{10} for further
development and use of this notation .

\vspace{2mm}


In the \textbf{first} part of our communicate umbral extensions of
the Stirling numbers of the both kinds are considered  and the
resulting new type  of Dobinski-like formulas are discovered [10].
These extensions naturally encompass the well known $q$-extensions
. The fact that $q$-extended Stirling numbers giving rise to the
umbral $q$-extended Dobinski formula interpreted as the average of
powers of random variable $X_q$ with the $q$-Poisson distribution
and are equivalent by re-scaling with the other Comtet -like
$q$-extended Stirling numbers [10] -  singles out the
$q$-extensions itself which appear to be a kind of bifurcation
point in the domain of umbral extensions . The further consecutive
umbral extensions of Carlitz-Gould $q$-Stirling numbers are
therefore realized in [10] in a two-fold way.

\vspace{2mm}

In the \textbf{second } part of our communicate one displays
\cite{11} how and  why Morgan Ward solution \cite{6} of $\psi$-
{\em difference calculus} nonhomogeneous equation
$\Delta_{\psi}f=\varphi$ in the form
$$f(x)=\sum_{n \geq
1}\frac{B_{n}}{n_{\psi}!}\varphi^{(n-1)}(x)+\int_{\psi}\varphi(x)
+p(x)$$ recently proposed by the present author \cite{12}- extends
here now to $\psi$- Appell polynomials case - almost {\em
automatically}. Illustrative specifications to $q$-calculus case
and Fibonomial calculus case \cite{2,13,14} were already made
explicit in \cite{12} exactly due to the of upside down notation
for objects of the  \textbf{EFOC }.

\vspace{2mm}





\subsection*{The First Part - Stirling numbers extensions and Dobinski-like formulae}

In this part of our communicate we follow [10] and we refer Reader
for further details to consult [10]. At the start let us recall
that the \textbf{two} standard \cite{15,16,17} $q$-extensions
Stirling numbers of the second kind  might be defined as follows:

\begin{equation}\label{eq4}
x_q^n=\sum_{k=0}^{n}\Big\{{n \atop k}\Big\}_q  x_q^{\underline k},
\end{equation}

where $ x_q=\frac{1-q^x}{1-q}$ and $x_q^{\underline
k}=x_q(x-1)_q...(x-k+1)_q $ , which corresponds to the $\psi$
sequence choice in the $q$-Gauss form
$\langle\frac{1}{n_q!}\rangle_{n\geq 0}$

\vspace{2mm}

and $q^\sim$-Stirling numbers

\begin{equation}\label{eq5}
x^n=\sum_{k=0}^{n}\Big\{{n \atop k}\Big\}^\sim_q \chi_{\underline
k}(x)
\end{equation}

where $\chi_{\underline k}(x)= x(x-1_q)(x-2_q)...(x-[k-1]_q)$


For these two classical by now $q$-extensions of  Stirling numbers
of the second kind  - the "$q$-standard" recurrences hold
respectively:

$$ \Big\{{{n+1}\atop k}\Big\}_q =
\sum_{l=0}^{n}\binom{n}{l}_q q^l\Big\{{l\atop {k-1}}\Big\}_q ;
n\geq 0 , k\geq 1,$$

$$ \Big\{{{n+1}\atop k}\Big\}^\sim_q =
\sum_{l=0}^{n}\binom{n}{l}_q q^{l-k+1}\Big\{{l\atop
{k-1}}\Big\}^\sim_q  ;  n\geq 0 , k\geq 1.$$

From the above it follows immediately that corresponding
$q$-extensions of  $B_n$ Bell numbers satisfy respective
recurrences:


$$ B_q(n+1) =
\sum_{l=0}^{n}\binom{n}{l}_q q^l B_q(l) ; n\geq 0 ,$$

$$ B^\sim_q(n+1) =
\sum_{l=0}^{n}\binom{n}{l}_q q^{l-k+1} \overline {B} ^\sim_q(l)  ;
n\geq 0 $$ where
$$ \overline {B}^\sim_q(l)=
\sum_{k=0}^{l} q^k\Big\{{l\atop k}\Big\}^\sim_q .
$$

\textbf{Note} that both definitions via (4) and (5) equations
consequently correspond to different $q$-counting   \cite{16} .\\

For applications to coherent state phisics see [17]  and
references therein.\\

 With any other choice out of countless choices of the
$\psi$ sequence the equation (5) becomes the definition of
$\psi^\sim$- Stirling (vide "Fibonomial-Stirling") numbers of the
second kind$\Big\{{n \atop k}\Big\}^\sim_\psi$  and then
$\psi^\sim$-Bell numbers $B^\sim_n(\psi)$ are defined as sums as
usual - where now $\chi_{\underline k}(x)$ in (5) is to be
replaced by $\psi_{\underline k}(x)=
x(x-1_\psi)(x-2_\psi)...(x-[k-1]_\psi)$. These $\psi^\sim$-
Stirling numbers of the second kind recognized for  $q$  case
properly as Comtet numbers in Wagner`s terminology \cite{16}
satisfy familiar recursion and are given by familiar formulas to
be presented soon.\\


The extension of definition (4) of the $q$- Stirling numbers of
the second kind beyond this $q$-case i.e. beyond the
$\psi=\langle\frac{1}{n_q!}\rangle_{n\geq 0}$ choice is not that
mnemonic and the " behavior " of the naturally  expected recursion
under extension  comprises quite a surprise (see: appendix 2.2 in
[10]).\\

Therefore  further consecutive umbral extension of Carlitz-Gould
$q$-Stirling numbers   $\Big\{{n\atop k}\Big\}_q $  and
$\Big\{{n\atop k}\Big\}^\sim_q $   is realized two-fold way [10].

The first "easy way" consists in almost mnemonic sometimes
replacement of $q$ subscript by $\psi$ because we learn from
\cite{16} that the equation (5) defines as a matter of fact the
specific case of Comtet numbers \cite{16} i.e. we define
$\psi$-extended Comtet-Stirling numbers of the second kind as
follows:


\begin{equation}\label{eq6}
x^n=\sum_{k=0}^{n}\Big\{{n \atop k}\Big\}^\sim_{\psi}
\psi_{\underline k}(x)
\end{equation}

where $\psi_{\underline k}(x)=
x(x-1_{\psi})(x-2_{\psi})...(x-[k-1]_{\psi}).$

As a consequence we have "for granted" :


\begin{equation}\label{eq7}
\Big\{{{n+1}\atop k}\Big\}^\sim_{\psi} = \Big\{{n\atop
{k-1}}\Big\}^\sim_{\psi} + k_{\psi}\Big\{{n\atop
k}\Big\}^\sim_{\psi} ;\quad n\geq 0 , k\geq 1 ;
\end{equation}

where \quad $\Big\{{n\atop 0}\Big\}^\sim_{\psi}=
\delta_{n,0},\quad\Big\{{n\atop k}\Big\}^\sim_{\psi}=0 ,\quad k>n
;\quad $\quad and the easy derivable  recurrence relations for
ordinary generating function now read

\vspace{2mm}

\begin{equation}\label{eq8}
G^\sim_{k_{\psi}}(x)=\frac{x}{1-k_{\psi}}G^\sim_{k_{\psi}-1}(x) ,
\quad k\geq 1
\end{equation}

where
$$G^\sim_{k_{\psi}}(x)= \sum_{n\geq 0}\Big\{{n\atop
k}\Big\}^\sim_{\psi}x^n     ,\quad k\geq 1 $$

\vspace{2mm}

from where as a consequence we arrive to what follows:
\vspace{5mm}

\begin{equation}\label{eq9}
G^\sim_{k_{\psi}}(x)=\frac{x^k}{(1-1_{\psi}x)(1-2_{\psi}x)...(1-k_{\psi}x)}
\quad, \quad k\geq 0
\end{equation}

\vspace{2mm}

and adapting reasoning from  \cite{18} we derive the following
explicit formula

\begin{equation}\label{eq10}
\Big\{{n\atop k}\Big\}^\sim_{\psi} = \frac
{r_{\psi}^n}{k_{\psi}!}\sum_{r=1}^{k}(-1)^{k-r}\binom{k_{\psi}}{r_{\psi}};
\quad n,k\geq 0 .
\end{equation}

\vspace{2mm}

Expanding the right hand side of (9) results in another explicit
formula for these $\psi$-case Comtet numbers i.e. we have

\vspace{2mm}

\begin{equation}\label{eq11}
\Big\{{n\atop k}\Big\}^\sim_{\psi} = \sum_{1\leq i_1 \leq
i_2\leq...\leq i_{n-k}\leq
k}(i_1)_{\psi}(i_2)_{\psi}...(i_{n-k})_{\psi}; \quad n,k\geq 0 .
\end{equation}

\vspace{2mm}

or equivalently  (compare with [16])

 \vspace{2mm}
\begin{equation}\label{eq12}
\Big\{{n\atop k}\Big\}^\sim_{\psi}=\sum_{d_1+ d_2+...+d_k =
n-k,\quad d_i\geq
0}1_{\psi}^{d_1}2_{\psi}^{d_2}...k_{\psi}^{d_k};\quad n,k\geq 0 .
\end{equation}

\vspace{2mm}

With help of $\psi^\sim$-\textbf{Stirling numbers} of the second
kind being defined equivalently by (6) , (7), (11)  or (12) we
define now  $\psi^\sim$-\textbf{Bell numbers} in a standard way
$$ B^\sim_n(\psi)=\sum_{k=0}^{n} \Big\{{n\atop
k}\Big\}^\sim_{\psi} ,\qquad n\geq 0 .$$

The recurrence  for  $B^\sim_n(\psi)$ is already quite involved
and complicated for the  $q$-extension case (see: the first
section in [10])- and no acceptable readable form of recurrence
for the $\psi$-extension case is known to us.

Nevertheless after adapting the  the corresponding Wilf`s
reasoning from \cite{18} we derive  for two variable ordinary
generating function for $\Big\{{n\atop k}\Big\}^\sim_{\psi}$
Stirling numbers of the second kind and the $\psi$-exponential
generating function for $ B^\sim_n(\psi)$  Bell numbers the
following  formulae

\begin{equation}\label{eq13}
C^\sim_{\psi}(x,y) = \sum_{n\geq 0} A^\sim_n (\psi,y)x^n ,
\end{equation}
where the $\psi$- exponential-like polynomials $ A^\sim_n
(\psi,y)$
$$ A^\sim_n (\psi,y)=\sum_{k=0}^{n} \Big\{{n\atop
k}\Big\}^\sim_{\psi}y^k$$ do satisfy the recurrence

$$ A^\sim_n (\psi,y)=  [y(1+\partial_{\psi}]A^\sim_{n-1}(\psi,y) \qquad n\geq 1 ,$$

hence
$$ A^\sim_n (\psi,y)=  [y(1+\partial_{\psi}]^n 1,\quad \qquad n\geq 0 ,$$
where  the linear operator $\partial_{\psi}$ acting on the algebra
of formal power series is being called (see: [1,2,3,4,5] and
references therein) the "$\psi$-derivative" and $\partial_{\psi}
y^n = n_{\psi}y^{n-1}.$

\vspace{2mm}

The $\psi$-exponential generating function $ F^{\sim}_{\psi}(x)=
\sum_{n\geq 0}B^\sim_n(\psi)\frac{x^n}{n_{\psi}!} $

for $ F^{\sim}_n(\psi)$ Bell numbers - again - after cautious
adaptation of the method from the Wilf`s generatingfunctionology
book  [18] we get only a little bit involved formula

\begin{equation}\label{eq14}
B^{\sim}_{\psi}(x)= \sum_{r\geq 0}\epsilon(\psi,r)
\frac{e_{\psi}[r_{\psi}x]}{r_{\psi}!}
\end{equation}

where (see: [6,7,1,2,3,4])
$$e_{\psi}(x) =
\sum_{n\geq 0}\frac{x^n}{n_{\psi}!}$$

while
\begin{equation}\label{eq15}
\epsilon(\psi,r)=\sum_{k=r}^{\infty} \frac{(-1)^{k-r}}{(k_{\psi}-
r_{\psi})!}
\end{equation}


and  for  $\psi$-extension the Dobinski like formula here now
reads

\begin{equation}\label{eq16}
B^{\sim}_n (\psi)= \sum_{r\geq 0}\epsilon(\psi,r)
\frac{r_{\psi}^n}{r_{\psi}!}.
\end{equation}

In the case of  Gauss $q$-extended  choice of
$\langle\frac{1}{n_q!}\rangle_{n\geq 0}$ admissible sequence of
extended umbral operator calculus  we have then

\begin{equation}\label{eq17}
\epsilon(q,r)=\sum_{k=r}^{\infty}
\frac{(-1)^{k-r}}{(k-r)_q!}q^{-\binom {r}{2}}
\end{equation}
and the new $q^\sim$-Dobinski  formula is given by

\begin{equation}\label{eq18}
B^{\sim}_n (\psi)= \sum_{r\geq 0}\epsilon(\psi,r)
\frac{r_{\psi}^n}{r_{\psi}!}.
\end{equation}
which for $ q=1$ becomes the Dobinski formula from 1887 [19].\\

As for the problem of how eventually one might  interpret the
$\psi^\sim$-Dobinski formulae (16) and (18) in the Rota-like way
see: [10].\\

In [10] you  find also details in support of  our conviction that
$q$-extensions seem  appear as a kind of bifurcation point in the
domain of umbral extensions.\\

For the discussion of the other way to arrive eventually to
another type of Dobinski formula  - via an attempt to
$\psi$-extend the equation (4) - we again refer the Reader to [10]
as this is quite a longer story with a surprise.


\vspace{2mm}


The parallel treatment  of the Comtet $ \left[ {{\begin{array}{*{20}c} {n}\\
{k}\end{array}} } \right]^\sim_\psi$ Stirling numbers of the first
kind is now not difficult.

Namely we define  as in [10] the $\psi^\sim$ Stirling numbers of
the first kind as the coefficients in the following expansion


\begin{equation}\label{eq19}
\psi_{\underline k}(x)=\sum_{r=0}^{k}\left[ {{\begin{array}{*{20}c} {k}\\
{r}\end{array}} } \right]^\sim_\psi x^r
\end{equation}

where -  recall $\psi_{\underline k}(x)=
x(x-1_{\psi})(x-2_{\psi})...(x-[k-1]_{\psi});\quad$ From the above
we then get

\begin{equation}\label{eq20}
\sum_{r=0}^{k}\left[ {{\begin{array}{*{20}c} {k}\\
{r}\end{array}} } \right]^\sim_\psi \Big\{{r\atop
l}\Big\}^\sim_{\psi}= \delta_{k,l}.
\end{equation}

Another (expected Whithney numbers of the first kind) $\psi^c$-
Stirling numbers of the first kind [10]  are defined as follows:


\begin{equation}\label{eq21}
\psi_{\overline k}(x)=\sum_{r=0}^{k}\left[ {{\begin{array}{*{20}c} {k}\\
{r}\end{array}} } \right]^c_\psi x^r
\end{equation}

where -  now  $\psi_{\overline k}(x)=
x(x+1_{\psi})(x+2_{\psi})...(x+[k-1]_{\psi});\quad$ More on that -
see [10].

\subsection*{The Second  Part -$\psi$- Appell polynomials` solutions of
the $Q(\partial_{\psi})$- difference  nonhomogeneous equation}


The second part is based on [11,12] to which we refer for more
details.

\vspace{2mm}

At first  recall \cite{3,1} the simple fact to be used in what
follows.

\begin{prop}
$Q(\partial_{\psi})$ is a $\psi$- delta operator iff there exists
invertible $S\in \Sigma _{\psi}$ such that
$Q(\partial_{\psi})=\partial_{\psi}S$.
\end{prop}


Formally: "$S=Q/\partial_{\psi}$" or "$S^{-1}=\partial_{\psi}/Q$".
In the sequel we use this abbreviation $Q(\partial_{\psi})\equiv
Q$.

$\psi$- {\em Appell} or generalized {\em Appell polynomials}
$\left\{A_{n}(x)\right\}_{n \geq 0}$ are defined according to
\begin{equation} \label{psi-A-1}
\partial_{\psi}A_{n}(x)=n_{\psi}A_{n-1}(x)
\end{equation}
and they naturally do satisfy the $\psi$ - Sheffer-Appell identity
\cite{3,1}
\begin{equation}\label{psi-A-2}
A_{n}(x+_{\psi}y)=\sum_{s=0}^{n}\binom{n}{s}
_{\psi}A_{s}(y)x^{n-s}.
\end{equation}
$\psi$-{\em Appell} or generalized {\em Appell polynomials}
$\left\{A_{n}(x)\right\}_{n \geq 0}$ are equivalently
characterized via their $\psi$- exponential generating function
\begin{equation}\label{psi-A-3}
\sum_{n \geq
0}z^{n}\frac{A_{n}(x)}{n_{\psi}!}=A(z)\exp_{\psi}\left\{xz\right\},
\end{equation}
where $A(z)$ is a formal series with constant term different from
zero - here normalized to one.

The $\psi$- exponential function of $\psi$-Appell-Ward numbers
$A_{n}=A_{n}(0)$ is
\begin{equation} \label{psi-A-4}
\sum_{n \geq 0}z^{n}\frac{A_{n}}{n_{\psi}!}=A(z).
\end{equation}
Naturally $\psi$- {\em Appell} $\left\{A_{n}(x)\right\}_{n \geq
0}$ satisfy the $\psi$- {\em difference} equation
\begin{equation}\label{psi-A-5}
QA_{n}(x)=n_{\psi}x^{n-1};\;\;\;'n\geq 0,
\end{equation}
because
$QA_{n}(x)=QS^{-1}x^{n}=Q(\partial_{\psi}/Q)x^{n}=\partial_{\psi}x^{n}=n_{\psi}x^{n-1}\;
;n \geq 0$. Therefore they play the same role in
$Q(\partial_{\psi})$- {\em difference} calculus as Bernoulli
polynomials do in standard difference calculus or $\psi$-{\em
Bernoulli-Ward polynomials} (see Theorem 16.1 in \cite{1} and
consult also \cite{12}) in $\psi$-{\em difference} calculus due to
the following: The central problem of the $Q(\partial_{\psi})$ -
{\em difference calculus} is:
$$ Q(\partial_{\psi})f=\varphi\;\;\;\;\;\;\;\;\;\;\varphi=?,$$
where $f, \varphi$ - are for example formal series or polynomials.

The idea of finding solutions is the $\psi$-{\em Finite Operator
Calculus} \cite{1,3,4,2} standard. As one knows  \cite{1,3}) any
$\psi$- delta operator $Q$ is of the form

$$Q(\partial_{\psi})=\partial_{\psi}S $$

where$ S\in \Sigma_{\psi}.$ Let $Q(\partial_{\psi})=\sum_{k \geq
1}\frac{q_{k}}{k_{\psi}!}\partial_{\psi}^{k},\;\;q_{1}\neq 0$.
Consider then $Q(\partial_{\psi}=\partial_{\psi}S)$ with
$S=\sum_{k \geq
0}\frac{q_{k+1}}{(k+1)_{\psi}!}\partial_{\psi}^{k}\equiv \sum_{k
\geq
0}\frac{s_{k}}{k_{\psi}!}\partial_{\psi}^{k};\;\;\;s_{0}=q_{1}\neq
0$.

We thus have for $S^{-1}\equiv\hat{A}$ - call it: $\psi$- {\em
Appell operator} - the obvious expression

$$\hat{A}\equiv S^{-1}=\frac{\partial_{\psi}}{Q_{\psi}}=\sum_{n\geq 0}\frac{A_{n}}{n_{\psi}!}\partial_{\psi}^{n}.$$

Now multiply the equation \quad $Q(\partial_{\psi})f=\varphi
$\quad  by $\hat{A}\equiv \sum_{n\geq
0}\frac{A_{n}}{n_{\psi}!}\partial_{\psi}^{n}$ thus getting



\begin{equation}\label{psi_A_6}
\partial_{\psi}f=\sum_{n\geq
0}\frac{A_{n}}{n_{\psi}!}\varphi^{(n)},\;\;\;\varphi^{(n)}=\partial_{\psi}\varphi^{(n-1)}.
\end{equation}
The solution then reads:


\begin{equation}\label{psi-A-7}
f(x)=\sum_{n \geq
1}\frac{A_{n}}{n_{\psi}!}\varphi^{(n-1)}(x)+\int_{\psi}\varphi(x)+p(x),
\end{equation}


where $p$ is "$Q(\partial_{\psi})$- periodic" i.e.
$Q(\partial_{\psi})p=0$. Compare with  \cite{12}

for "$+_{\psi}1$- periodic" i.e. $p(x+_{\psi}1)=p(x)$ i.e.
$\Delta_{\psi}p=0$. Here
 the relevant  $\psi$ - integration $\int_{\psi}\varphi(x)$ is defined as in
\cite{1}. We recall it in brief. Let us introduce the following
representation for $\partial_{\psi}$ "difference-ization"


$$ \partial_{\psi}=\hat{n}_{\psi}\partial_{0}\; ;\;\;\;\hat{n}_{\psi}x^{n-1}=n_{\psi}x^{n-1};\;\;n\geq 1,$$


where $\partial_{0}x^{n}=x^{n-1}$ i.e. $\partial_{0}$ is the $q=0$
Jackson derivative. $\partial_{0}$ is identical with divided
difference operator. Then we define the linear mapping
$\int_{\psi}$ accordingly:

$$ \int_{\psi}x^{n}=\left( \hat{x}\frac{1}{\hat{n}_{\psi}}\right)x^{n}=\frac{1}{(n+1)_{\psi}}x^{n+1};\;\;\;n\geq 0$$
where of course  $\partial_{\psi}\circ \int_{\psi}=id$.


\section{Examples}
\renewcommand{\labelenumi}{(\alph{enumi})}
\begin{enumerate}

\item The case of $\psi$- {\em Bernoulli-Ward} polynomials and
$\Delta_{\psi}$- {\em difference calculus} was considered in
detail in \cite{12} following \cite{6}. \item Specification of (a)
to the Gauss and Heine originating $q$-umbral calculus case
\cite{6,3,4,5,2} was already presented in \cite{12}.

\item Specification of (a) to the Lucas originating FFOC - case
was also presented in \cite{12} (here: FFOC={\bf F}ibonomial {\bf
F}inite {\bf O}perator {\bf C}alculus), see example 2.1 in
\cite{2}). Recall: the {\em Fibonomial coefficients} -already
known in 19-th century to Lucas [20]-  are defined "binomially" as

$$\binom{n}{k}_{F}=\frac{F_{n}!}{F_{k}!F_{n-k}!}=\binom{n}{n-k}_{F},$$


where ($F_{n}$- {\em Fibonacci numbers} in up-side down notation:
$n_{F}\equiv F_{n}\neq
0$,\\$n_{F}!=n_{F}(n-1)_{F}(n-2)_{F}(n-3)_{F}\ldots
2_{F}1_{F};\;\;0_{F}!=1$;\\$n_{F}^{\underline{k}}=n_{F}(n-1)_{F}\ldots
(n-k+1)_{F};\;\;\;\binom{n}{k}_{F}\equiv
\frac{n^{\underline{k}}_{F}}{k_{F}!}$.

\vspace{2mm}

We shall call the corresponding linear difference operator
$\partial_{F};\;\;\partial_{F}x^{n}=n_{F}x^{n-1};\;\;n\geq 0$ the
$F$-derivative. Then in conformity with \cite{6} and with notation
as in \cite{1}-\cite{13,14} one has:


$$E^{a}(\partial_{F})=\sum_{n\geq 0}\frac{a^{n}}{n_{F}!}\partial_{F}^{n}$$


for the corresponding [3,1,4] generalized translation operator
$E^{a}(\partial_{F})$. The $\psi$- integration for the moment is
still not explored $F$- integration and we arrive at the
\textbf{$F$-
Bernoulli} polynomials unknown till now.\\

{\bf Note:} recently a combinatorial interpretation of Fibonomial
coefficient has been found \cite{14} by the present author.

\item The other examples of $Q(\partial_{\psi})$- {\em difference
calculus} - expected naturally to be of primary importance in
applications are provided by the possible use of such $\psi$-
Appell polynomials as:

\begin{itemize}
\item

$\psi$-Hermite polynomials $\left\{H_{n,\psi}\right\}_{n\geq 0}$:


$$ H_{n,\psi}(x)=\left[\sum_{k\geq 0}\left(-\frac{1}{2}\right)^{k}
\frac{\partial_{\psi}^{2k}}{k_{\psi}!}\right]x^{n}\;\;\;n\geq 0;$$


\item

$\psi$ - Laguerre polynomials $\left\{ L_{n,\psi}\right\}_{n\geq
0}$ \cite{3}:

\begin{multline*}
L_{n,q}(x) = \frac{{n_{q}} }{{n}}\hat {x}_{\psi}
\left[{\frac{{1}}{{\partial _{\psi} - 1}}}\right]^{-n}x^{n-1} =
\frac{{n_{\psi}} }{{n}}\hat {x}_{\psi} \left( {\partial _{\psi} -
1} \right)^{n}x^{n-1} = \\ = \frac{{n_{\psi}} }{{n}}\hat
{x}_{\psi} \sum\limits_{k = 1}^{n} \left( { - 1} \right)^{k}\left(
{{\begin{array}{*{20}c}
 {n} \hfill \\
 {k} \hfill \\
\end{array}} } \right) \partial _{\psi} ^{n - k}x^{n - 1} =\\
=\frac{{n_{\psi}} }{{n}}\sum\limits_{k = 1}^{n} {} \left( { - 1}
\right)^{k}\left( {{\begin{array}{*{20}c}
 {n} \hfill \\
 {k} \hfill \\
\end{array}} } \right) \left( {n - 1} \right)_{\psi} ^{\underline {n - k}
}\frac{{k}}{{k_{\psi}} }x^{k} .
\end{multline*}


For $q=1$ in $q$-extended case one recovers the known formula :

\begin{center}
$L_{n,q=1}(x) = \sum\limits_{k = 1}^{n} {} \left( { - 1}
\right)^{k}\frac{{n_{q} !}}{{k_{q} !}}\left(
{{\begin{array}{*{20}c}
 {n - 1} \hfill \\
 {k - 1} \hfill \\
\end{array}} } \right)x^{k}$.
\end{center}

\end{itemize}
\end{enumerate}



\begin{thebibliography}{99}
\parskip 0pt

\bibitem{1}
A.K.Kwa\'sniewski {\it Main  theorems of extended finite operator
calculus} Integral Transforms and Special Functions, {\bf 14} No 6
(2003): 499-516

\bibitem{2}
A.K.Kwa\'sniewski {\it On Simple Characterizations of Sheffer
$psi$-polynomials and Related Propositions of the Calculus of
Sequences} ,Bulletin de la Soc. des Sciences et de Lettres de
Lodz,  {\bf 52}  Ser. Rech. Deform.  36 (2002):45-65, ArXiv:
math.CO/0312397 .

\bibitem{3}
A.K.Kwa\'sniewski {\it Towards  $\psi$-extension of Finite
Operator Calculus of Rota} Rep. Math. Phys. {\bf 48} No3
(2001):305-342

\bibitem{4}
A.K.Kwa\'sniewski{\it  On extended finite operator calculus of
Rota and quantum groups}  Integral Transforms and Special
Functions {\bf 2} No4 (2001):333-340

\bibitem{5}
A.K.Kwa\'sniewski {\it First contact remarks on umbra difference
calculus references streams}   Bulletin de la Soc. des Sciences et
de Lettres de Lodz,  in press;  ArXiv: math.CO/0403139 v1  8 March
2004

\bibitem{6}
M. Ward: {\em A calculus of sequences}, Amer.J.Math. Vol.58,
1936,pp.255-266

\bibitem{7}
O.V. Viskov:   {\it Operator characterization of  generalized
Appel polynomials } Soviet Math. Dokl. {\bf 16}, 1521 (1975).

\bibitem{8}
G.~Markowsky, {\it Differential operator and theory of binomial
enumeration}, Math. Anal. Appl. {\bf 63} (1978), 145--155.


\bibitem{9}
G.-C. Rota: {\em Finite Operator Calculus\/}, Academic Press, New
York 1975.

\bibitem{10} A.K.Kwa\'sniewski {\it On umbral extensions  of Stirling numbers
and Dobinski-like formulas} submitted for publication
ArXiv:math.CO/0411002 30 October 2004

\bibitem{11}
A.K.Kwa\'sniewski {\it $\psi$-Appell polynomials` solutions of an
umbral difference nonhomogeneous equation} Bulletin de la Societe
des Sciences et des Lettres de Lodz (54) Serie: Recherches sur les
Deformations {\bf  45} (2004), 11-15. ArXiv: math.CO/0405578 30
May 2004

\bibitem{12}
A. K. Kwa\'sniewski: {\it On basic Bernoulli-Ward polynomials },
Bulletin de la Societe des Sciences et des Lettres de Lodz (54)
Serie: Recherches sur les Deformations {\bf 45} (2004), 5-10.
ArXiv:math.CO/0405577 30 May 2004


\bibitem{13}
A.K.Kwa\'sniewski   Fibonomial cumulative connection constants
Bulletin of the ICA {\bf 44} (2005), 81-92. ArXiv: math.CO/0406006
1 Jun 2004


\bibitem{14}
A.K.Kwa\'sniewski: More on combinatorial interpretation of the
fibonomial coefficients  Bulletin de la Soc. des Sciences et de
Lettres de Lodz (54) Serie: Recherches sur les Deformations {\bf
44} (2004),23-38.  ArXiv: math.CO/0402344 v1 22 Feb 2004
\bibitem{15}


S.C. Milne {\it A $q$-analog of restricted growth functions,
Dobinski's equality, and Charlier polynomials} ,Trans. Amer. Math.
Soc. {\bf 245} (1978) 89-118

\bibitem{16}
Carl G. Wagner {\it    Partition Statistics and q-Bell Numbers (q
= -1)}, Journal of Integer Sequences {\bf 7}(2004) Article 04.1.1

\bibitem{17}
J.Katriel, M. Kibler {\it Normal ordering for deformed boson
operators and operator-valued deformed Stirling numbers} J. Phys.
A: Math. Gen. {\bf 25} (1992): 2683-26-91

\bibitem{18}
H.S.Wilf {\it Generatingfunctionology} Boston: Academic Press,
1990

\bibitem{19}
G. Dobinski {\it Summierung der Reihe S .... f\"{u}r m = 1, 2, 3,
4, 5, ....} Grunert Archiv (Arch. Math. Phys) {\bf 61}, 333-336,
(1877)

\bibitem{20}
E. Lucas, {\it Th\'eorie des fonctions num\'eriques simplement
p\'eriodiques}, American Journal of Mathematics {\bf 1} (1878):
184--240

\end{thebibliography}
\end{document}

math0411145.extracted_bib
\begin{thebibliography}{99}
\parskip 0pt

\bibitem{1}
A.K.Kwa\'sniewski {\it Main  theorems of extended finite operator
calculus} Integral Transforms and Special Functions, {\bf 14} No 6
(2003): 499-516

\bibitem{2}
A.K.Kwa\'sniewski {\it On Simple Characterizations of Sheffer
$psi$-polynomials and Related Propositions of the Calculus of
Sequences} ,Bulletin de la Soc. des Sciences et de Lettres de
Lodz,  {\bf 52}  Ser. Rech. Deform.  36 (2002):45-65, ArXiv:
math.CO/0312397 .

\bibitem{3}
A.K.Kwa\'sniewski {\it Towards  $\psi$-extension of Finite
Operator Calculus of Rota} Rep. Math. Phys. {\bf 48} No3
(2001):305-342

\bibitem{4}
A.K.Kwa\'sniewski{\it  On extended finite operator calculus of
Rota and quantum groups}  Integral Transforms and Special
Functions {\bf 2} No4 (2001):333-340

\bibitem{5}
A.K.Kwa\'sniewski {\it First contact remarks on umbra difference
calculus references streams}   Bulletin de la Soc. des Sciences et
de Lettres de Lodz,  in press;  ArXiv: math.CO/0403139 v1  8 March
2004

\bibitem{6}
M. Ward: {\em A calculus of sequences}, Amer.J.Math. Vol.58,
1936,pp.255-266

\bibitem{7}
O.V. Viskov:   {\it Operator characterization of  generalized
Appel polynomials } Soviet Math. Dokl. {\bf 16}, 1521 (1975).

\bibitem{8}
G.~Markowsky, {\it Differential operator and theory of binomial
enumeration}, Math. Anal. Appl. {\bf 63} (1978), 145--155.


\bibitem{9}
G.-C. Rota: {\em Finite Operator Calculus\/}, Academic Press, New
York 1975.

\bibitem{10} A.K.Kwa\'sniewski {\it On umbral extensions  of Stirling numbers
and Dobinski-like formulas} submitted for publication
ArXiv:math.CO/0411002 30 October 2004

\bibitem{11}
A.K.Kwa\'sniewski {\it $\psi$-Appell polynomials` solutions of an
umbral difference nonhomogeneous equation} Bulletin de la Societe
des Sciences et des Lettres de Lodz (54) Serie: Recherches sur les
Deformations {\bf  45} (2004), 11-15. ArXiv: math.CO/0405578 30
May 2004

\bibitem{12}
A. K. Kwa\'sniewski: {\it On basic Bernoulli-Ward polynomials },
Bulletin de la Societe des Sciences et des Lettres de Lodz (54)
Serie: Recherches sur les Deformations {\bf 45} (2004), 5-10.
ArXiv:math.CO/0405577 30 May 2004


\bibitem{13}
A.K.Kwa\'sniewski   Fibonomial cumulative connection constants
Bulletin of the ICA {\bf 44} (2005), 81-92. ArXiv: math.CO/0406006
1 Jun 2004


\bibitem{14}
A.K.Kwa\'sniewski: More on combinatorial interpretation of the
fibonomial coefficients  Bulletin de la Soc. des Sciences et de
Lettres de Lodz (54) Serie: Recherches sur les Deformations {\bf
44} (2004),23-38.  ArXiv: math.CO/0402344 v1 22 Feb 2004
\bibitem{15}


S.C. Milne {\it A $q$-analog of restricted growth functions,
Dobinski's equality, and Charlier polynomials} ,Trans. Amer. Math.
Soc. {\bf 245} (1978) 89-118

\bibitem{16}
Carl G. Wagner {\it    Partition Statistics and q-Bell Numbers (q
= -1)}, Journal of Integer Sequences {\bf 7}(2004) Article 04.1.1

\bibitem{17}
J.Katriel, M. Kibler {\it Normal ordering for deformed boson
operators and operator-valued deformed Stirling numbers} J. Phys.
A: Math. Gen. {\bf 25} (1992): 2683-26-91

\bibitem{18}
H.S.Wilf {\it Generatingfunctionology} Boston: Academic Press,
1990

\bibitem{19}
G. Dobinski {\it Summierung der Reihe S .... f\"{u}r m = 1, 2, 3,
4, 5, ....} Grunert Archiv (Arch. Math. Phys) {\bf 61}, 333-336,
(1877)

\bibitem{20}
E. Lucas, {\it Th\'eorie des fonctions num\'eriques simplement
p\'eriodiques}, American Journal of Mathematics {\bf 1} (1878):
184--240

\end{thebibliography}