# A unified Generating function of the $q$-Genocchi polynomials with Their
Interpolation Functions

Serkan Aracı  University of Gaziantep, Faculty of Science and Arts, Department
of Mathematics, 27310 Gaziantep, TURKEY<EMAIL_ADDRESS>,  Mehmet Açıkgöz
University of Gaziantep, Faculty of Science and Arts, Department of
Mathematics, 27310 Gaziantep, TURKEY<EMAIL_ADDRESS>,  Hassan Jolany
School of Mathematics, Statistics and Computer Science, University of Tehran,
Iran<EMAIL_ADDRESS>and  Jong Jin Seo  Department of Applied
Mathematics, Pukyong National University, Busan, 608-737,Korea
<EMAIL_ADDRESS>
(Date: April 18, 2011)

###### Abstract.

The purpose of this paper is to construct of the unification $q$-extension
Genocchi polynomials. We give some interesting relations of this type of
polynomials. Finally, we derive the $q$-extensions of Hurwitz-zeta type
functions from the Mellin transformation of this generating function which
interpolates the unification of $q$-extension of Genocchi polynomials.

###### Key words and phrases:

Genocchi numbers and polynomials, $q$-Genocchi numbers and polynomials

###### 1991 Mathematics Subject Classification:

05A10, 11B65, 28B99, 11B68, 11B73.

##  1\. Introduction, Definitions and Notations

Recently, many mathematician have studied to unification Bernoulli, Genocchi,
Euler and Bernstein polynomials (see [20,21]). Ozden [20] introduced $p$-adic
distribution of the unification of the Bernoulli, Euler and Genocchi
polynomials and derived some properties of this type of unification
polynomials.

In [20], Ozden constructed the following generating function:

(1.1)
$\sum_{n=0}^{\infty}y_{n,\beta}\left(x;k,a,b\right)\frac{t^{n}}{n!}=\frac{2^{1-k}t^{k}e^{xt}}{\beta^{b}e^{t}-a^{b}},\text{
}\left|t+b\ln\left(\frac{\beta}{a}\right)\right|<2\pi$

where $k\in\mathbb{N}=\left\\{1,2,3,...\right\\},$ $a,b\in\mathbb{R}$ and
$\beta\in\mathbb{C}.$ The polynomials $y_{n,\beta}\left(x;k,a,b\right)$ are
the unification of the Bernoulli, Euler and Genocchi polynomials.

Ozden showed, $\beta=b=1,$ $k=0$ and $a=-1$ into (1.1), we have

$y_{n,1}\left(x;0,-1,1\right)=E_{n}\left(x\right),$

where $E_{n}\left(x\right)$ denotes classical Euler polynomials, which are
defined by the following generating function:

(1.2)
$\frac{2e^{xt}}{e^{t}+1}=\sum_{n=0}^{\infty}E_{n}\left(x\right)\frac{t^{n}}{n!},\text{
}\left|t\right|<\pi,\text{ }$

In [5,20], classical Genocchi polynomials defined as follows:

(1.3)
$\frac{2te^{xt}}{e^{t}+1}=\sum_{n=0}^{\infty}G_{n}\left(x\right)\frac{t^{n}}{n!},\text{
}\left|t\right|<\pi,\text{ }$

From (1.2) and (1.3), we easily see,

(1.4) $G_{n}\left(x\right)=nE_{n-1}\left(x\right),$

For a fixed real number $\left|q\right|<1,$ we use the notation of $q$-number
as

$\left[x\right]_{q}=\frac{1-q^{x}}{1-q}\text{, \ \ (see [1-4,6-24]),}$

Thus, we note that $\lim_{q\rightarrow 1}\left[x\right]_{q}=x$.

In [1,7,8,10], $q$-extension of Genocchi polynomials are defined as follows:

$G_{n+1,q}\left(x\right)=\left(n+1\right)\left[2\right]_{q}\sum_{l=0}^{\infty}\left(-1\right)^{l}q^{l}\left[x+l\right]_{q}^{n}.$

In [2], $\left(h,q\right)$-extension of Genocchi polynomials are defined as
follows:

$G_{n+1,q}^{\left(h\right)}\left(x\right)=\left(n+1\right)\left[2\right]_{q}\sum_{l=0}^{\infty}\left(-1\right)^{l}q^{\left(h-1\right)l}\left[x+l\right]_{q}^{n}.$

In this paper, we shall construct unification of $q$-extension of the Genocchi
polynomials, however we shall give some interesting relationships. Moreover,
we shall derive the $q$-extension of Hurwitz-zeta type functions from the
Mellin transformation of this generating function which interpolates.

##  2\.  Novel Generating Functions of $q$-extension of Genocchi polynomials

######  Definition 1.

Let $a,b\in\mathbb{R}$ , $\beta\in\mathbb{C}$ and
$k\in\mathbb{N}=\left\\{1,2,3,...\right\\},$Then the unification of
$q$-extension of Genocchi polynomials defined as follows:

(2.1)
$\tciFourier_{\beta,q}\left(t,x\left|k,a,b\right.\right)=\sum_{n=0}^{\infty}S_{n,\beta,q}\left(x\left|k,a,b\right.\right)\frac{t^{n}}{n!}$

and

(2.2)
$\tciFourier_{\beta,q}\left(t,x\left|k,a,b\right.\right)=-\left[2\right]_{q}^{1-k}t^{k}\sum_{m=0}^{\infty}\beta^{bm}a^{-bm-b}e^{\left[m+x\right]_{q}t}.$

where into (2.1) substituting $x=0,$
$S_{n,\beta,q}\left(0\left|k,a,b\right.\right)=S_{n,\beta,q}\left(k,a,b\right)$
are called unification of $q$-extension of Genocchi numbers.

As well as, from (2.1) and (2.2) Ozden’s constructed the following generating
function, namely, we obtain (1.1),

$\lim_{q\rightarrow
1}\tciFourier_{\beta,q}\left(t,x\left|k,a,b\right.\right)=\frac{2^{1-k}t^{k}e^{xt}}{\beta^{b}e^{t}-a^{b}}\text{.}$

By (2.2), we see readily,

(2.3)
$\displaystyle\sum_{n=0}^{\infty}S_{n,\beta,q}\left(x\left|k,a,b\right.\right)\frac{t^{n}}{n!}$
$\displaystyle=$
$\displaystyle-\left[2\right]_{q}^{1-k}t^{k}\sum_{m=0}^{\infty}\beta^{bm}a^{-bm-b}e^{\left[m+x\right]_{q}t}$
$\displaystyle=$
$\displaystyle\frac{e^{\left[x\right]_{q}t}}{q^{kx}}\left(-\left[2\right]_{q}^{1-k}\left(q^{x}t\right)^{k}\sum_{m=0}^{\infty}\beta^{bm}a^{-bm-b}e^{\left(q^{x}t\right)\left[m\right]_{q}}\right)$
$\displaystyle=$ $\displaystyle
q^{-kx}\left(\sum_{n=0}^{\infty}\left[x\right]_{q}^{n}\frac{t^{n}}{n!}\right)\left(\sum_{n=0}^{\infty}q^{nx}S_{n,\beta,q}\left(k,a,b\right)\frac{t^{n}}{n!}\right)$

From (2.3) by using Cauchy product we get

$\sum_{n=0}^{\infty}S_{n,\beta,q}\left(x\left|k,a,b\right.\right)\frac{t^{n}}{n!}=q^{-kx}\sum_{n=0}^{\infty}\left(\sum_{l=0}^{n}\binom{n}{l}q^{lx}S_{l,\beta,q}\left(k,a,b\right)\left[x\right]_{q}^{n-l}\right)\frac{t^{n}}{n!}$

By comparing coefficients of $\frac{t^{n}}{n!}$ in the both sides of the above
equation, we arrive at the following theorem:

######  Theorem 1.

For $a,b\in\mathbb{R}$ , $\beta\in\mathbb{C}$ which $k$ is
$\mathop{\mathrm{p}ositive}$ integer. We obtain

$S_{n,\beta,q}\left(x\left|k,a,b\right.\right)=\sum_{l=0}^{n}\binom{n}{l}q^{lx}S_{l,\beta,q}\left(k,a,b\right)\left[x\right]_{q}^{n-l}$

As well as, we obtain corollary 1:

######  Corollary 1.

For $a,b\in\mathbb{R}$ , $\beta\in\mathbb{C}$ which $k$ is
$\mathop{\mathrm{p}ositive}$ integer. We obtain,

(2.4)
$S_{n,\beta,q}\left(x\left|k,a,b\right.\right)=\left(S_{\beta,q}\left(k,a,b\right)+\left[x\right]_{q}\right)^{n}$

with usual the convention about replacing
$\left(S_{\beta,q}\left(x\left|k,a,b\right.\right)\right)^{n}$by
$S_{n,\beta,q}\left(x\left|k,a,b\right.\right).$

By applying the definition of generating function of
$S_{n,\beta,q}\left(x\left|k,a,b\right.\right),$ we have

$\displaystyle\sum_{n=0}^{\infty}S_{n,\beta,q}\left(x\left|k,a,b\right.\right)\frac{t^{n}}{n!}$
$\displaystyle=$
$\displaystyle-\left[2\right]_{q}^{1-k}t^{k}\sum_{m=0}^{\infty}\beta^{bm}a^{-bm-b}\left(\sum_{n=0}^{\infty}\left[m+x\right]_{q}^{n}\frac{t^{n}}{n!}\right)$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\left(-\left[2\right]_{q}^{1-k}\sum_{m=0}^{\infty}\beta^{bm}a^{-bm-b}\left[m+x\right]_{q}^{n}\right)\frac{t^{n+k}}{n!}$

So we derive the Theorem 2 which we state hear:

######  Theorem 2.

For $a,b\in\mathbb{R}$ , $\beta\in\mathbb{C}$ which $k$ is positive integer.
We obtain,

(2.5)
$S_{n,\beta,q}\left(x\left|k,a,b\right.\right)=-\frac{n!\left[2\right]_{q}^{1-k}}{\left(n-k\right)!}\sum_{m=0}^{\infty}\beta^{bm}a^{-bm-b}\left[m+x\right]_{q}^{n-k}$

With regard to (2.5), we see after some calculations

(2.6) $\displaystyle S_{n,\beta,q}\left(x\left|k,a,b\right.\right)$
$\displaystyle=$
$\displaystyle-\frac{n!\left[2\right]_{q}^{1-k}}{a^{b}\left(n-k\right)!}\left(\frac{1}{1-q}\right)^{n-k}\sum_{l=0}^{n-k}\binom{n-k}{l}\left(-1\right)^{l}q^{lx}\sum_{m=0}^{\infty}\left(\frac{\beta^{b}}{a^{b}}\right)^{m}q^{lm}$
$\displaystyle=$
$\displaystyle\frac{n!\left[2\right]_{q}^{1-k}}{a^{b}\left(n-k\right)!}\left(\frac{1}{1-q}\right)^{n-k}\sum_{l=0}^{n-k}\binom{n-k}{l}\left(-1\right)^{l}q^{lx}\frac{1}{\beta^{b}q^{l}-a^{b}}$
$\displaystyle=$
$\displaystyle\frac{k!\left[2\right]_{q}^{1-k}}{a^{b}}\left(\frac{1}{1-q}\right)^{n-k}\sum_{l=0}^{n-k}\binom{n}{k}\binom{n-k}{l}\left(-1\right)^{l}q^{lx}\frac{1}{\beta^{b}q^{l}-a^{b}}$

From (2.6) and well known identity
$\left[\binom{n}{k}\binom{n-k}{l}=\binom{n}{k+l}\binom{k+l}{k}\right],$ we
obtain the following theorem:

######  Theorem 3.

For $a,b\in\mathbb{R}$ , $\beta\in\mathbb{C}$ which $k$ is
$\mathop{\mathrm{p}ositive}$ integer. We obtain

(2.7)
$S_{n,\beta,q}\left(x\left|k,a,b\right.\right)=\frac{k!\left[2\right]_{q}^{1-k}}{a^{b}}\left(\frac{1}{1-q}\right)^{n-k}\sum_{l=k}^{n}\binom{n}{l}\binom{l}{k}\left(-1\right)^{l-k}q^{\left(l-k\right)x}\frac{1}{\beta^{b}q^{l-k}-a^{b}}$

We put $x\rightarrow 1-x,\beta\rightarrow\beta^{-1},q\rightarrow q^{-1}$ and
$a\rightarrow a^{-1}$ into (2.7), namely,

$\displaystyle S_{n,\beta^{-1},q^{-1}}\left(1-x\left|k,a^{-1},b\right.\right)$
$\displaystyle=$
$\displaystyle\frac{k!\left[2\right]_{q^{-1}}^{1-k}}{a^{-b}}\left(\frac{1}{1-q^{-1}}\right)^{n-k}\sum_{l=k}^{n}\binom{n}{l}\binom{l}{k}\left(-1\right)^{l-k}q^{-\left(l-k\right)\left(1-x\right)}\frac{1}{\beta^{-b}q^{-\left(l-k\right)}-a^{-b}}$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n-k}q^{k-1}q^{n-k}k!\left[2\right]_{q}^{1-k}a^{b}\left(\frac{1}{1-q}\right)^{n-k}\sum_{l=k}^{n}\binom{n}{l}\binom{l}{k}\left(-1\right)^{l-k}q^{k-l}q^{\left(l-k\right)x}\frac{\beta^{b}q^{l-k}a^{b}}{a^{b}-\beta^{b}q^{l-k}}$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n-k-1}q^{n-1}a^{3b}\beta^{b}\left(\frac{k!\left[2\right]_{q}^{1-k}}{a^{b}}\left(\frac{1}{1-q}\right)^{n-k}\sum_{l=k}^{n}\binom{n}{l}\binom{l}{k}\left(-1\right)^{l-k}q^{x\left(l-k\right)}\frac{1}{\beta^{b}q^{l-k}-a^{b}}\right)$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n-k-1}q^{n-1}a^{3b}\beta^{b}S_{n,\beta,q}\left(x\left|k,a,b\right.\right)$

So, we obtain symmetric properties of
$S_{n,\beta,q}\left(x\left|k,a,b\right.\right)$ as follows:

######  Theorem 4.

For $a,b\in\mathbb{R}$ , $\beta\in\mathbb{C}$ which $k$ is
$\mathop{\mathrm{p}ositive}$ integer. We obtain

$S_{n,\beta^{-1},q^{-1}}\left(1-x\left|k,a^{-1},b\right.\right)=\left(-1\right)^{n-k-1}q^{n-1}a^{3b}\beta^{b}S_{n,\beta,q}\left(x\left|k,a,b\right.\right).$

By (2.2), we see

(2.8)
$\displaystyle\frac{\beta}{a}\tciFourier_{\beta,q}\left(t,1\left|k,a,b\right.\right)-\tciFourier_{\beta,q}\left(t,0\left|k,a,b\right.\right)$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\left(\left(\frac{\beta}{a}\right)S_{n,\beta,q}\left(1\left|k,a,b\right.\right)-S_{n,\beta,q}\left(k,a,b\right)\right)\frac{t^{n}}{n!}=-\frac{\left[2\right]_{q}^{1-k}}{a^{b}}t^{k}$

From (2.8), we obtain the following theorem:

######  Theorem 5.

For $a,b\in\mathbb{R}$ , $\beta\in\mathbb{C}$ which $k$ is
$\mathop{\mathrm{p}ositive}$ integer. We obtain

(2.9)
$S_{n,\beta,q}\left(k,a,b\right)-\left(\frac{\beta}{a}\right)S_{n,\beta,q}\left(1\left|k,a,b\right.\right)=\left\\{\QATOP{0,\text{
\ \ }n\neq k}{\frac{\left[2\right]_{q}^{1-k}}{a^{b}}k!,n=k}\right.$

From (2.4) and (2.9), we obtain corollary as follows:

######  Corollary 2.

For $a,b\in\mathbb{R}$ , $\beta\in\mathbb{C}$, which is $k$ positive integer.
We get

$S_{n,\beta,q}\left(k,a,b\right)-\frac{\beta}{aq^{k}}\left(qS_{\beta,q}\left(k,a,b\right)+1\right)^{n}=\left\\{\QATOP{0,n\neq
k}{\frac{\left[2\right]_{q}^{1-k}}{a^{b}}k!,n=k}\right.$

with the usual convention about replacing
$\left(S_{\beta,q}\left(k,a,b\right)\right)^{n}$ by
$S_{n,\beta,q}\left(k,a,b\right).$

From $\left(9\right),$ now, we shall obtain distribution relation for
unification $q$-extension of Genocchi polynomials, after some calculations,
namely,

$\displaystyle S_{n,\beta,q}\left(x\left|k,a,b\right.\right)$ $\displaystyle=$
$\displaystyle-\frac{n!\left[2\right]_{q}^{1-k}}{a^{b}\left(n-k\right)!}\sum_{m=0}^{\infty}\left(\frac{\beta}{a}\right)^{bm}\left[m+x\right]_{q}^{n-k}$
$\displaystyle=$
$\displaystyle-\frac{n!\left[2\right]_{q}^{1-k}}{a^{b}\left(n-k\right)!}\sum_{m=0}^{\infty}\sum_{l=0}^{d-1}\left(\frac{\beta}{a}\right)^{b\left(l+md\right)}\left[l+md+x\right]_{q}^{n-k}$
$\displaystyle=$
$\displaystyle\left[d\right]_{q}^{n-k}\sum_{l=0}^{d-1}\left(\frac{\beta}{a}\right)^{bl}\left(-\frac{n!\left[2\right]_{q}^{1-k}}{a^{b}\left(n-k\right)!}\sum_{m=0}^{\infty}\left(\frac{\beta^{d}}{a^{d}}\right)^{bm}\left[m+\frac{x+l}{d}\right]_{q^{d}}^{n-k}\right)$
$\displaystyle=$
$\displaystyle\frac{\left[2\right]_{q}^{1-k}}{\left[2\right]_{q^{d}}^{1-k}}\left[d\right]_{q}^{n-k}\sum_{l=0}^{d-1}\left(\frac{\beta}{a}\right)^{bl}S_{n,\beta^{d},q^{d}}\left(\frac{x+l}{d}\left|k,a^{d},b\right.\right)$

Therefore, we obtain the following theorem:

######  Theorem 6.

$\left(distribution\text{ \ }formula\text{ }for\text{
}S_{n,\beta,q}\left(x\left|k,a,b\right.\right)\right)$For $a,b\in\mathbb{R}$ ,
$\beta\in\mathbb{C}$ which $k$ is $\mathop{\mathrm{p}ositive}$ integer. We
obtain,

$S_{n,\beta,q}\left(x\left|k,a,b\right.\right)=\frac{\left[2\right]_{q}^{1-k}}{\left[2\right]_{q^{d}}^{1-k}}\left[d\right]_{q}^{n-k}\sum_{l=0}^{d-1}\left(\frac{\beta}{a}\right)^{bl}S_{n,\beta^{d},q^{d}}\left(\frac{x+l}{d}\left|k,a^{d},b\right.\right)$

##  3\. Interpolation function of the polynomials
$S_{n,\beta,q}\left(x\left|k,a,b\right.\right)$

In this section, we give interpolation function of the generating functions of
$S_{n,\beta,q}\left(x\left|k,a,b\right.\right)$ however, this function is
meromorphic function. This function interpolates
$S_{n,\beta,q}\left(x\left|k,a,b\right.\right)$ at negative integers.

For $s\in\mathbb{C}$ , by applying the Mellin transformation to (2.2), we
obtain

$\displaystyle\boldsymbol{\Im}_{\beta,q}\left(s;x,a,b\right)$ $\displaystyle=$
$\displaystyle\frac{\left(-1\right)^{k+1}}{\Gamma\left(s\right)}\mathop{\displaystyle\oint}t^{s-k-1}\tciFourier_{\beta,q}\left(-t,x\left|k,a,b\right.\right)dt$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}^{1-k}\sum_{m=0}^{\infty}\beta^{bm}a^{-bm-b}\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}t^{s-1}e^{-t\left[m+x\right]_{q}t}$

So, we have

$\boldsymbol{\Im}_{\beta,q}\left(s;x,a,b\right)=\left[2\right]_{q}^{1-k}\sum_{m=0}^{\infty}\frac{\beta^{bm}a^{-bm-b}}{\left[m+x\right]_{q}^{s}}$

We define $q-$extension Hurwitz-zeta type function as follows theorem:

######  Theorem 7.

For $a,b\in\mathbb{R}$ , $\beta,s\in\mathbb{C}$ which $k$ is
$\mathop{\mathrm{p}ositive}$ integer. We obtain,

(3.1)
$\boldsymbol{\Im}_{\beta,q}\left(s;x,a,b\right)=\left[2\right]_{q}^{1-k}\sum_{m=0}^{\infty}\frac{\beta^{bm}a^{-bm-b}}{\left[m+x\right]_{q}^{s}}$

for all $s\in\mathbb{C}.$ We note that
$\boldsymbol{\Im}_{\beta,q}\left(s;x,a,b\right)$ is analytic function in the
whole complex $s$-plane.

######  Theorem 8.

For $a,b\in\mathbb{R}$ , $\beta\in\mathbb{C}$ which $k$ is
$\mathop{\mathrm{p}ositive}$ integer. We obtain,

$\boldsymbol{\Im}_{\beta,q}\left(-n;x,a,b\right)=-\frac{\left(n-k\right)!}{n!}S_{n,\beta,q}\left(x\left|k,a,b\right.\right).$

###### Proof.

Let $a,b\in\mathbb{R}$ , $\beta\in\mathbb{C}$ and $k\in\mathbb{N}$ with
$k\in\mathbb{N}=\left\\{1,2,3,...\right\\}$. $\Gamma\left(s\right)$, has
simple poles at $z=-n=0,-1,-2,-3,\cdots.$ The residue of
$\Gamma\left(s\right)$ is

$\mathop{\mathrm{R}e}s\left(\Gamma\left(s\right),-n\right)=\frac{\left(-1\right)^{n}}{n!}.$

We put $s\rightarrow-n$ into (3.1) and using the above relations, the desired
result can be obtained.

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