---
abstract: 'The article considers the generalized $k$-Bessel functions and represents it as Wright functions. Then we study the monotonicity properties of the ratio of two different orders $k$- Bessel functions, and the ratio of the $k$-Bessel and the $m$-Bessel functions. The log-convexity with respect to the order of the $k$-Bessel also given. An investigation regarding the monotonicity of the ratio of the $k$-Bessel and $k$-confluent hypergeometric functions are discussed.'
address:
- |
    Department of Mathematics\
    King Faisal University, Al Ahsa 31982, Saudi Arabia
- |
    Department of Mathematics\
    Prince Sattam bin Abdulaziz University, Saudi Arabia
author:
- 'Saiful. R. Mondal'
- 'Kottakkaran S. Nisar'
title: 'Inequalities for the modified $k$- Bessel function'
---

Introduction {#Intro}
============

One of the generalization of the classical gamma function $\Gamma$ studied in [@Diaz] is defined by the limit formula $$\begin{aligned}
\label{eqn-1}
\Gamma_k(x) := \lim_{n \to \infty} \frac{n! \; k^n (n^k)^{\tfrac{x}{k}-1}}{(x)_{n, k}}, \quad k>0,\end{aligned}$$ where $(x)_{n, k}:=x(x+k) (x+2k)\ldots (x+(n-1)k)$ is called $k$-Pochhammer symbol. The above $k-$gamma function also have an integral representation as $$\begin{aligned}
\label{eqn-2}
\Gamma_k(x)= \int_0^\infty t^{x-1} e^{-\frac{t^{k}}{k}} dt, \quad \operatorname{Re}(x)>0.\end{aligned}$$ Properties of the $k$-gamma functions have been studies by many researchers [@CGK; @CGK2; @VK; @MM; @Mubeen13]. Follwoing properties are required in sequel:

-   $\Gamma _{k}\left( x+k\right) =x\Gamma _{k}\left( x\right)$

-   $\Gamma _{k}\left( x\right) =k^{\frac{x}{k}-1}\Gamma \left( \frac{x}{k}\right)$

-   $\Gamma _{k}\left( k\right)=1$

-   $\Gamma _{k}\left( x+nk\right)=\Gamma _{k}(x) (x)_{n, k}$

Motivated with the above generalization of the $k$-gamma functions, Romero et. al.[@Romero-Cerutti] introduced the $k-$Bessel function of the first kind defined by the series $$\begin{aligned}
\label{k1}
J_{k,\nu }^{\gamma ,\lambda }\left( x\right) :=\sum_{n=0}^{\infty }\frac{\left( \gamma \right) _{n,\;k}}{\Gamma _{k}\left( \lambda n+\upsilon +1\right)
}\frac{\left( -1\right) ^{n}\left( x/2\right) ^{n}}{\left( n!\right) ^{2}},\end{aligned}$$where $k\in \mathbb{R^{+}}$; $\alpha,\lambda,\gamma,\upsilon \in C$; $\operatorname{Re}(\lambda)>0$ and $\operatorname{Re}(\upsilon) >0$. They also established two recurrence relations for $J_{k,\nu }^{\gamma ,\lambda }$.

In this article, we are considering the following function: $$\begin{aligned}
\label{eqn-modfb}
I_{k,\nu }^{\gamma ,\lambda }\left( x\right) :=\sum_{n=0}^{\infty }\frac{\left( \gamma \right) _{n,\;k}}{\Gamma _{k}\left( \lambda n+\upsilon +1\right)
}\frac{\left( x/2\right) ^{n}}{\left( n!\right) ^{2}},\end{aligned}$$ Since $$\lim_{k, \lambda, \gamma \to 1 }I_{k,\nu }^{\gamma ,\lambda }\left( x\right)=
\sum_{n=0}^{\infty }\frac{1}{\Gamma\left( n+\upsilon +1\right)
}\frac{\left( x/2\right) ^{n}}{n!}=\left(\frac{2}{x}\right)^{\frac{\nu}{2}} I_\nu(\sqrt{2x}),$$ the classical modified Bessel functions of first kind. In this sense, we can call $I_{k,\nu }^{\gamma ,\lambda }$ as the modified $k$-Bessel functions of first kind. In fact, we can express both $J_{k,\nu }^{\gamma ,\lambda }$ and $I_{k,\nu }^{\gamma ,\lambda }$ together in $$\label{eqn-genb}
\mathtt{W}_{k,\nu, c }^{\gamma,\lambda }(x) :=\sum_{n=0}^{\infty}\frac{
( \gamma )_{n,\;k}}{\Gamma_{k}(\lambda n+ \nu +1)
}\frac{(-c)^n( x/2) ^{n}}{\left( n!\right) ^{2}}, \quad c \in \mathbb{R}.$$ We can termed $\mathtt{W}_{k,\nu }^{\gamma ,\lambda }$ as the generalized $k$-Bessel function.

First we study the representation formulas for $\mathtt{W}_{k,\nu }^{\gamma ,\lambda }$ in term of the classical Wright functions. Then we will study about the monotonicity and log-convexity properties of $I_{k,\nu }^{\gamma ,\lambda }$.

Representation formula for the generalized $k$-Bessel function {#sec1}
==============================================================

The generalized hypergeometric function ${}_pF_q(a_1,\ldots, a_p;c_1,\ldots,c_q;x)$, is given by the power series $$\label{eqn:gen-hyp-func}
{}_pF_q(a_1,\ldots, a_p;c_1,\ldots,c_q;z) = \sum_{k=0}^{\infty}\dfrac{(a_1)_{k}
\cdots (a_p)_{k}}{(c_1)_{k}\cdots(c_q)_{k}(1)_{k}}z^k, \quad \quad |z|<1,$$ where the $c_{i}$ can not be zero or a negative integer. Here $p$ or $q$ or both are allowed to be zero. The series $(\ref{eqn:gen-hyp-func})$ is absolutely convergent for all finite $z$ if $p\leq q$ and for $|z|<1$ if $p=q+1$. When $p>q+1$, then the series diverge for $z \not=0$ and the series does not terminate.

The generalized Wright hypergeometric function ${}_p\psi_q(z)$ is given by the series $$\label{eqn-9-bessel}
{}_p\psi_q(z)={}_p\psi_q\left[\begin{array}{c}
(a_i,\alpha_i)_{1,p} \\
(b_j,\beta_j)_{1,q}
\end{array}\bigg|z\right]=\displaystyle\sum_{k=0}^{\infty}\frac{\prod_{i=1}^{p}\Gamma(a_{i}+\alpha_{i}k)}
{\prod_{j=1}^{q}\Gamma(b_{j}+\beta_{j}k)}
\frac{z^{k}}{k!},$$ where $a_i, b_j\in \mathbb{C}$, and real $\alpha_i, \beta_j\in \mathbb{R}$ ($i=1,2,\ldots,p; j =1,2,\ldots,q$). The asymptotic behavior of this function for large values of argument of $z\in \mathbb{C}$ were studied in [@CFox; @Kilbas] and under the condition $$\label{eqn-10-bessel}
\displaystyle\sum_{j=1}^{q}\beta_{j}-\displaystyle\sum_{i=1}^{p}\alpha_{i}>-1$$ in literature [@Wright-2; @Wright-3]. The more properties of the Wright function are investigated in [@Kilbas; @Kilbas-itsf; @KST].

Now we will give the representation of the generalized $k$-Bessel functions in terms of the Wright and generalized hypergeometric functions.

Let, $k \in \mathbb{R}$ and $\lambda ,\gamma ,\nu \in \mathbb{C}$ such that $\operatorname{Re}(
\lambda) >0,\operatorname{Re}( \nu ) >0.$ Then $$\mathtt{W}_{k,\nu,c }^{\gamma ,\lambda }(x) =\frac{1}{k^{\frac{\nu+k+1}{k}}\Gamma\left(\frac{\gamma}{k}\right)}
{}_1\psi_2\left[\begin{array}{ccc}
\left(\frac{\gamma}{k}, 1\right)& \\
\left(\frac{\nu+1}{k}, \frac{\gamma}{k}\right) & (1, 1)
\end{array}\bigg|-\frac{c x}{2 k^{\frac{\lambda}{k}-1}}\right]$$

Using the relations $\Gamma _{k}\left( x\right) =k^{\frac{x}{k}-1}\Gamma \left( \frac{x}{k}\right)$ and $\Gamma _{k}\left( x+nk\right)=\Gamma _{k}(x) (x)_{n, k}$, the generalized $k$-Bessel functions defined in $\eqref{eqn-genb}$ can be rewrite as $$\begin{aligned}
\mathtt{W}_{k,\nu,c }^{\gamma ,\lambda }(x)
&=\sum_{n=0}^\infty \frac{\Gamma_k(\gamma+n k)}{\Gamma_k(\lambda n+\nu+1) \Gamma_k(\gamma)} \frac{(-c)^n}{(n!)^2} \left(\frac{x}{2}\right)^n\\
&=\frac{1}{k^{\frac{\nu+k+1}{k}} \Gamma\left(\frac{\gamma}{k}\right)}\sum_{n=0}^\infty \frac{\Gamma\left(\frac{\gamma}{k}+n \right)}{\Gamma\left(\frac{\lambda}{k} n+\frac{\nu+1}{k}\right) \Gamma\left(\frac{\gamma}{k}\right)} \frac{(-c)^n}{\Gamma(n+1)\Gamma(n+1)} \left(\frac{x}{2 k^{\frac{\lambda}{k}-1}}\right)^n\\
&=\frac{1}{k^{\frac{\nu+k+1}{k}}\Gamma\left(\frac{\gamma}{k}\right)}
{}_1\psi_2\left[\begin{array}{ccc}
\left(\frac{\gamma}{k}, 1\right)& \\
\left(\frac{\nu+1}{k}, \frac{\gamma}{k}\right) & (1, 1)
\end{array}\bigg|-\frac{c x}{2 k^{\frac{\lambda}{k}-1}}\right]\end{aligned}$$ Hence the result follows.

Monotonicty and log-convexity properties {#sec2}
========================================

This section discuss the monotonicity and log-convexity properties for the modified $k$-Bessel functions $\mathtt{W}_{k,\nu,-1 }^{\gamma ,\lambda }(x)=\mathtt{I}_{k,\nu}^{\gamma ,\lambda }(x)$.

Following lemma due to Biernacki and Krzyż [@Biernacki-Krzy] will be required.

\[lemma:1\][@Biernacki-Krzy] Consider the power series $f(x)=\sum_{k=0}^\infty a_k x^k$ and $g(x)=\sum_{k=0}^\infty b_k x^k$, where $a_k \in \mathbb{R}$ and $b_k > 0$ for all $k$. Further suppose that both series converge on $|x|<r$. If the sequence $\{a_k/b_k\}_{k\geq 0}$ is increasing (or decreasing), then the function $x \mapsto f(x)/g(x)$ is also increasing (or decreasing) on $(0,r)$.

The above lemma still holds when both $f$ and $g$ are even, or both are odd functions.

The following results holds true for the modified $k$-Bessel functions.

1.  For $\mu \geq \nu>-1$, the function $x \mapsto \mathtt{I}_{k,\mu}^{\gamma ,\lambda }(x)/\mathtt{I}_{k,\nu}^{\gamma ,\lambda }(x)$ is increasing on $(0, \infty)$ for some fixed $k >0$.

2.  If $k\geq \lambda \geq m>0$, the function $x \mapsto \mathtt{I}_{k,\nu}^{\gamma ,\lambda }(x)/\mathtt{I}_{m,\nu}^{\gamma ,\lambda }(x)$ is increasing on $(0, \infty)$ for some fixed $\nu >-1$ and $\gamma \geq \nu+1$.

3.  The function $\nu \mapsto \mathcal{I}_{k,\nu}^{\gamma ,\lambda }(x)$ is log-convex on $(0, \infty)$ for some fixed $k, \gamma>0$ and $x>0$. Here, $\mathcal{I}_{k,\nu}^{\gamma ,\lambda }(x):=\Gamma_k(\nu+1)\mathtt{I}_{k,\nu}^{\gamma ,\lambda }(x)$.

4.  Suppose that $\lambda \geq k>0$ and $\nu>-1$. Then

    1.  The function $x \mapsto \mathtt{I}_{k,\nu}^{\gamma ,\lambda }(x)/\Phi _{k}\left( a,c;x\right)$ is decreasing on $(0, \infty)$ for $a \geq c >0$ and $0<\gamma \leq \nu+1$. Here, $\Phi _{k}\left( a; c; x\right)$ is the $k$-confluent hypergeometric functions.

    2.  The function $x \mapsto \mathtt{I}_{k,\nu}^{\gamma ,\lambda }(x)/\Phi _{k}\left( \gamma; \lambda; x/2\right)$ is decreasing on $(0, 1)$ for $\gamma >0$ and $0< k \leq \lambda \leq \nu+1$.

    3.  The function $x \mapsto \mathtt{I}_{k,\nu}^{\gamma ,\lambda }(x)/\Phi _{k}\left( \gamma; \lambda; x/2\right)$ is decreasing on $[1, \infty)$ for $\gamma >0$ and $0< k \leq \min\{\lambda,\nu+1\}$.

${\bf (1)}$ Form it follows that $$\mathtt{I}_{ k, \nu}^{\gamma ,\lambda }(x)= \sum_{n=0}^\infty a_n(\nu) x^n\quad
\text{and} \quad \mathtt{I}_{ k, \nu}^{\gamma ,\lambda }(x)= \sum_{n=0}^\infty a_n(\mu) x^n,$$ where $$a_n(\nu)= \frac{(\gamma)_{n,k}}{\Gamma_k(\lambda n+\nu+1) (n!)^2 2^n}
\quad \text{and} \quad
a_n(\mu)= \frac{(\gamma)_{n,k}}{\Gamma_k(\lambda n+\mu+1) (n!)^2 2^n}$$ Consider the function $$f(t):= \frac{\Gamma_k(\lambda t+\mu+1)}{\Gamma_k(\lambda t+\nu+1)}.$$ Then the logarithmic differentiation yields $$\begin{aligned}
\frac{f'(t)}{f(t)}= \lambda( \Psi_k(\lambda t+\mu+1)-\Psi_k(\lambda t+\nu+1)).\end{aligned}$$ Here, $\Psi_k=\Gamma_k'/\Gamma_k $ is the $k$-digamma functions studied in [@Kwara14] and defined by $$\begin{aligned}
\label{def-digamma}
\Psi_k(t)=\frac{\log(k)-\gamma_1}{k}-\frac{1}{t}+\sum_{n=1}^\infty \frac{t}{nk(nk+t)}\end{aligned}$$ where $\gamma_1$ is the Euler-Mascheroni’s constant.

A calculation yields $$\begin{aligned}
\label{def-digamma-2} \Psi_k'(t)=\sum_{n=0}^\infty  \frac{1}{(nk+t)^2}, \quad k>0 \quad \text{and} \quad t>0.\end{aligned}$$ Clearly, $\Psi_k$ is increasing on $(0, \infty)$ and hence $f'(t)>0$ for all $t\geq0$ if $\mu \geq \nu>-1$. This, in particular, implies that the sequence $\{d_n\}_{n \geq 0}=\{a_n(\nu)/a_n(\mu)\}_{n \geq 0}$ is increasing and hence the conclusion follows from Lemma $\ref{lemma:1}$.

[**(2)**]{}. This result also follows from Lemma $\ref{lemma:1}$ if the sequence $\{d_n\}_{n \geq 0}=\{a_n^k(\nu)/a_n^m(\mu)\}_{n \geq 0}$ is increasing for $k \geq m >0$. Here, $$a_{n}^{k}\left( \nu \right) =\frac{\left( \gamma \right) _{n,k}}{\Gamma _{k}\left( \lambda n+\nu+1\right) \left( n!\right) ^{2}} \quad \text{and} \quad
a_{n}^{m}\left( \nu \right)  =\frac{\left( \gamma \right) _{n,m}}{\Gamma _{m}\left( \lambda n+\nu+1\right) \left( n!\right) ^{2}},$$ which together with the identity $\Gamma _{k}\left( x+nk\right)=\Gamma _{k}(x) (x)_{n, k}$ gives $$\begin{aligned}
d_n&=\frac{\left( \gamma \right) _{n,k}}{\left( \gamma \right) _{n,m}} \frac{
\Gamma _{m}\left( \lambda n+\nu+1\right) }{\Gamma _{k}\left( \lambda n+\nu+1\right)}\\
&= \frac{
\Gamma _{k}\left( \gamma +nk\right)\Gamma _{m}\left( \lambda n+\nu+1\right) }{\Gamma _{k}\left( \gamma +nm\right)\Gamma _{k}\left( \lambda n+\nu+1\right)}.\end{aligned}$$ Now to show that $\{d_n\}$ is increase, consider the function $$f(y):=\frac{
\Gamma _{k}\left( \gamma +yk\right)\Gamma _{m}\left( \lambda y+\nu+1\right) }{\Gamma _{k}\left( \gamma +ym\right)\Gamma _{k}\left( \lambda y+\nu+1\right)}$$ The logarithmic differentiation of $f$ yields $$\begin{aligned}
\label{3}
\frac{f'(y)}{f(y)}= k \Psi_k(\gamma +yk)+ \lambda \Psi_m\left( \lambda y+\nu+1\right)-m  \Psi_m(\gamma +ym )-\lambda \Psi_k\left( \lambda y+\nu+1\right)\end{aligned}$$ If $\gamma \geq  \nu+1$ and $k \geq \lambda \geq m $, then can be rewrite as $$\begin{aligned}
\label{44}
\frac{f'(y)}{f(y)}\geq  \lambda \big(\Psi_k(\nu+1 +yk)- \Psi_k\left( \lambda y+\nu+1\right)\big)+ m\big( \Psi_m\left( \lambda y+\nu+1\right)- \Psi_m(\nu+1 +ym )\big) \geq 0.\end{aligned}$$ This conclude that $f$, and consequently the sequence $\{d_n\}_{n\geq 0}$, is increasing. Finally the result follows from the Lemma \[lemma:1\].

[**(3).**]{} It is known that sum of the log-convex functions is log-convex. Thus, to prove the result it is enough to show that $$\nu \mapsto a_{n}^{k}\left( \nu \right) :=\frac{\left( \gamma \right) _{n,k}\Gamma _{k}\left( \nu+1\right)}{\Gamma _{k}\left( \lambda n+\nu+1\right) \left( n!\right) ^{2}}$$ is log-convex.

A logarithmic differentiation of $a_n(\nu)$ with respect to $\nu$ yields $$\begin{aligned}
\frac{\partial}{\partial \nu} \log\left(a_{n}^{k}\left( \nu \right)\right)=\Psi_k\left(\nu+1\right) - \Psi_k\left( \lambda n+\nu+1\right).\end{aligned}$$ This along with gives $$\begin{aligned}
\frac{\partial^2}{\partial\nu^2}\log\left(a_{n}^{k}\left( \nu \right)\right)
&=\Psi'_k\left(\nu+1\right) - \Psi'_k\left( \lambda n+\nu+1\right)\\
&=\sum_{r=0}^\infty  \frac{1}{(rk+\nu+1)^2} - \sum_{r=0}^\infty  \frac{1}{(rk+\lambda n+\nu+1)^2}\\
&=\sum_{r=0}^\infty  \frac{\lambda n(2 rk+\lambda n+2\nu+2)}{(rk+\nu+1)^2(rk+\lambda n+\nu+1)^2} >0,\end{aligned}$$ for all $n \geq 0$, $k >0$ and $\nu>-1$. Thus, $\nu \mapsto a_{n}^{k}\left( \nu \right)$ is log-convex and hence the conclusion.\
[**(4).**]{} Denote $\Phi _{k}\left( a,c;x\right)=\sum_{n=0}^\infty c_{n, k}( a, c) x^{n}$ and $\mathtt{I}_{ k, \nu}^{\gamma ,\lambda }(x)= \sum_{n=0}^\infty a_n(\nu) x^n,$ where $$a_n(\nu)= \frac{(\gamma)_{n,k}}{\Gamma_k(\lambda n+\nu+1) (n!)^2 2^n}\quad \text{and} \quad 
d_{n,k}\left( a,c\right) =\frac{\left( a\right) _{n,k}}{\left( c\right)
_{n,k}n!}$$with $v>-1$ and $a,c, \lambda, \gamma, k>0.$ To apply Lemma \[lemma:1\], consider the sequence $\left\{ w_{n}\right\} _{n\geq 0}$ defined by $$\begin{aligned}
w_{n} =\frac{a_{n}\left( \nu \right) }{d_{n, k}\left( a,c\right) }&=&\frac{\Gamma _{k}\left( \gamma +nk\right) }{2^{n}\Gamma _{k}\left( \gamma \right)
\Gamma _{k}\left( \lambda n+\alpha +1\right) \left( n!\right) ^{2}}.\frac{\Gamma _{k}\left( a\right) \Gamma _{k}\left( c+nk\right) n!}{\Gamma
_{k}\left( a+nk\right) \Gamma _{k}\left( c\right) } \\
&=&\frac{\Gamma _{k}\left( a\right) }{\Gamma _{k}\left( \gamma \right)
\Gamma _{k}\left( c\right) }\rho _{k}\left( n\right)\end{aligned}$$where $$\rho _{k}\left( x\right) =\frac{\Gamma _{k}\left( \gamma +xk\right) \Gamma
_{k}\left( c+xk\right) }{\Gamma _{k}\left( \lambda x+\nu +1\right) \Gamma
_{k}\left( a+xk\right) 2^x \Gamma(x+1) }.$$In view of the increasing properties of $\Psi_k$ on $(0, \infty)$, and $$\frac{\rho ^{\prime }\left( x\right) }{\rho \left( x\right) }= k \psi
_{k}\left( \gamma +xk\right) +k\psi _{k}\left( c+xk\right) -\lambda \psi _{k}\left(
\lambda x+\alpha +1\right) -k \psi _{k}\left( a+xk\right),$$ it follows that for $a\geq c>0$, $\lambda \geq k$ and $\nu+1\geq \gamma$, the function $\rho$ is decreasing on $
\left( 0,\infty \right) $ and thus the sequence $\left\{ w_{n}\right\} _{n\geq
0} $ also decreasing. Finally the conclusion for $(a)$ follows from the Lemma \[lemma:1\].

In the case $(b)$ and $(c)$, the sequence $\{w_n\}$ reduces to $$\begin{aligned}
w_{n} =\frac{a_{n}\left( \nu \right) }{d_{n, k}\left(\gamma, \lambda\right) }&=&\frac{\rho _{k}\left( n\right)}{\Gamma _{k}\left( \lambda \right)
 }\end{aligned}$$where $$\rho _{k}\left( x\right) =\frac{\Gamma _{k}\left( \lambda +xk\right) }{\Gamma _{k}(\nu +1+\lambda x) \Gamma(x+1) }.$$Now as in the proof of part (a) $$\frac{\rho _{k}'\left( x\right)}{\rho _{k}\left( x\right)} =k \Psi_k(\lambda +xk)-\lambda \Psi_k(\nu+1 +xk)-\Psi(x+1)>0,$$ if $\nu+1+ \lambda x \geq \lambda + xk$. Now for $x \in (0,1)$, this inequality holds if $0< k \leq \lambda \leq \nu+1$, while for $x \geq 1$, it is required that $k \leq \min\{\lambda, \nu+1\}.$

[999]{} L.G. Romero, G.A.Dorrego and R.A. Cerutti, The k-Bessel function of first kind, International Mathematical forum, 38(7)(2012), 1859–1854.

GN. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library Edition, Camdridge University Press, Camdridge (1995). Reprinted (1996)

A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher transcendental functions, I, II, McGraw-Hill Book Company, Inc., New York, 1953. NewYork, Toronto, London, 1953.

R. Diaz and E. Pariguan, On hypergeometric functions and k-Pochhammer symbol, Divulgaciones Matematicas 15(2) (2007), 179–192

Kwara Nantomah, Edward Prempeh, Some Inequalities for the k-Digamma Function, Mathematica Aeterna, 4(5) (2014), 521–525.

S. Mubeen, M. Naz and G. Rahman, A note on k-hypergemetric differential equations, Journal of Inequalities and Special Functions, 4(3) (2013), 8–43.

M. Biernacki and J. Krzyż, On the monotonity of certain functionals in the theory of analytic functions, Ann. Univ. Mariae Curie-Sk lodowska. Sect. A. 9 (1957), 135–147.

C. G. Kokologiannaki, Properties and inequalities of generalized $k$-gamma, beta and zeta functions, Int. J. Contemp. Math. Sci. 5(13-16) (2010), 653–660.

C. G. Kokologiannaki and V. Krasniqi, Some properties of the $k$-gamma function, Matematiche (Catania), 68(1) (2013), 13–22.

V. Krasniqi, A limit for the $k$-gamma and $k$-beta function, Int. Math. Forum, 5(33-36) (2010), 1613–1617.

M. Mansour, Determining the $k$-generalized gamma function $\Gamma\sb k(x)$ by functional equations, Int. J. Contemp. Math. Sci., 4(21-24) (2009), 1037–1042.

G. E. Andrews, R. Askey and R. Roy, Special functions, Cambridge Univ. Press, Cambridge, 1999.

C. Fox, The asymptotic expansion of generalized hypergeometric functions, Proc. London. Math. Soc., 27(4) (1928), 389–400.

A. A. Kilbas, M. Saigo and J. J. Trujillo, On the generalized Wright function, Fract. Calc. Appl. Anal. 5(4) (2002), 437–460.

A. A. Kilbas and N. Sebastian, Generalized fractional integration of Bessel function of the first kind, Integral Transforms Spec. Funct. 19 (11-12) (2008), 869–883.

A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204, Elsevier, Amsterdam, 2006.

E. D. Rainville, Special functions, Macmillan, New York, 1960.

E. M. Wright, The asymptotic expansion of integral functions defined by Taylor series, Philos. Trans. Roy. Soc. London, Ser. A. 238 (1940), 423–451.

E. M. Wright, The asymptotic expansion of the generalized hypergeometric function, Proc. London Math. Soc. (2) 46(1940), 389–408.
