A note on the $(h,q)$-zeta-type function with weight α

Abstract

The objective of this paper is to derive the symmetric property of an $(h,q)$-zeta function with weight α. By using this property, we give some interesting identities for $(h,q)$-Genocchi polynomials with weight α. As a result, our applications possess a number of interesting properties which we state in this paper.

MSC:11S80, 11B68.

1 Introduction

Recently, Kim has developed a new method by using the q-Volkenborn integral (or p-adic q-integral on ${\mathbb{Z}}_p$) and has added weight to q-Bernoulli numbers and polynomials and investigated their interesting properties (see [1]). He also showed that these polynomials are closely related to weighted q-Bernstein polynomials and derived novel properties of q-Bernoulli numbers with weight α by using the symmetric property of weighted q-Bernstein polynomials with the help of the q-Volkenborn integral (for more details, see [2]). Afterward, Araci et al. have introduced weighted $(h,q)$-Genocchi polynomials and defined $(h,q)$-zeta-type function with weight α by applying the Mellin transformation to the generating function of the $(h,q)$-Genocchi polynomials with weight α which interpolates for $(h,q)$-Genocchi polynomials with weight α at negative integers (for details, see [3]). In this paper, we also consider a $(h,q)$-zeta-type function with weight α and derive some interesting properties.

We firstly list some notations as follows.

Imagine that p is a fixed odd prime. Throughout this work, ℤ, ${\mathbb{Z}}_p$, ${\mathbb{Q}}_p$ and ${\mathbb{C}}_p$ will denote by the ring of integers, the field of p-adic rational numbers and the completion of the algebraic closure of ${\mathbb{Q}}_p$, respectively. Also, we denote ${\mathbb{N}}^{\ast }=\mathbb{N}\cup \{0\}$ and $exp(x)=e^x$. Let $v_p:{\mathbb{C}}_p\to \mathbb{Q}\cup \{\mathrm{\infty }\}$ (ℚ is the field of rational numbers) denote the p-adic valuation of ${\mathbb{C}}_p$ normalized so that $v_p(p)=1$. The absolute value on ${\mathbb{C}}_p$ will be denoted as $|\cdot |$, and $|x{|}_p=p^{-v_p(x)}$ for $x\in {\mathbb{C}}_p$. When one speaks of q-extensions, q is considered in many ways, e.g., as an indeterminate, a complex number $q\in \mathbb{C}$, or a p-adic number $q\in {\mathbb{C}}_p$. If $q\in \mathbb{C}$, we assume that $|q|<1$. If $q\in {\mathbb{C}}_p$, we assume $|1-q{|}_p<p^{-\frac{1}{p-1}}$ so that $q^x=exp(xlogq)$ for $|x{|}_p\le 1$. We use the following notation:

$${[x]}_q=\frac{1-q^x}{1-q},{[x]}_{-q}=\frac{1-{(-q)}^x}{1+q}.$$

(1.1)

We want to note that ${lim}_{q\to 1}{[x]}_q=x$; cf. [1–23].

For a fixed positive integer d, set

$$\begin{array}{c}X=X_d=\underset{\overleftarrow{n}}{lim}\mathbb{Z}/d{p}^n\mathbb{Z},\hfill \\ X^{\ast }=\underset{(a,p)=1}{\bigcup _{0<a<dp}}a+dp{\mathbb{Z}}_p\hfill \end{array}$$

and

$$a+d{p}^n{\mathbb{Z}}_p=\{x\in X\mid x\equiv a(modd{p}^n)\},$$

where $a\in \mathbb{Z}$ satisfies the condition $0\le a<d{p}^n$ (see [1–23]).

The following p-adic q-Haar distribution was defined by Kim:

$${\mu }_q(x+p^n{\mathbb{Z}}_p)=\frac{q^x}{{[p^n]}_q}$$

for any positive n (see [12, 13]).

Let $\mathit{UD}({\mathbb{Z}}_p)$ be the set of uniformly differentiable functions on ${\mathbb{Z}}_p$. We say that f is a uniformly differentiable function at a point $a\in {\mathbb{Z}}_p$ if the difference quotient

$$F_f(x,y)=\frac{f(x)-f(y)}{x-y}$$

has a limit $f^{\mathrm{\prime }}(a)$ as $(x,y)\to (a,a)$ and denote this by $f\in \mathit{UD}({\mathbb{Z}}_p)$. In [12] and [13], the p-adic q-integral of the function $f\in \mathit{UD}({\mathbb{Z}}_p)$ is defined by Kim as follows:

$$I_q(f)={\int }_{{\mathbb{Z}}_p}f(\xi )d{\mu }_q(\xi )=\underset{n\to \mathrm{\infty }}{lim}\sum _{\xi =0}^{p^n-1}f(\xi ){\mu }_q(\xi +p^n{\mathbb{Z}}_p).$$

(1.2)

The bosonic integral is considered as the bosonic limit $q\to 1$, $I_1(f)={lim}_{q\to 1}{I}_q(f)$. Similarly, the p-adic fermionic integration on ${\mathbb{Z}}_p$ is defined by Kim [8] as follows:

$$I_{-q}(f)=\underset{q\to -q}{lim}{I}_q(f)={\int }_{{\mathbb{Z}}_p}f(x)d{\mu }_{-q}(x).$$

(1.3)

By using a fermionic p-adic q-integral on ${\mathbb{Z}}_p$, $(h,q)$-Genocchi polynomials are defined by [3]

$$\begin{array}{rcl}\frac{{\tilde{G}}_{n+1,q}^{(\alpha ,h)}(x)}{n+1}& =& {\int }_{{\mathbb{Z}}_p}{q}^{(h-1)\xi }{[x+\xi ]}_{q^{\alpha }}^{n}d{\mu }_{-q}(\xi )\\ =& \underset{n\to \mathrm{\infty }}{lim}\frac{1}{{[p^n]}_{-q}}\sum _{\xi =0}^{p^n-1}{(-1)}^{\xi }{[x+\xi ]}_{q^{\alpha }}^n{q}^{h\xi }.\end{array}$$

(1.4)

For $x=0$ in (1.4), we have ${\tilde{G}}_{n,q}^{(\alpha ,h)}(0):={\tilde{G}}_{n,q}^{(\alpha ,h)}$ are called $(h,q)$-Genocchi numbers with weight α which is defined by

$${\tilde{G}}_{0,q}^{(\alpha ,h)}=0\text{and}{q}^h\frac{{\tilde{G}}_{m+1}^{(\alpha ,h)}(1)}{m+1}+\frac{{\tilde{G}}_{m+1}^{(\alpha ,h)}}{m+1}=\{\begin{array}{cc}{[2]}_q\hfill & \text{if }m=0,\hfill \\ 0\hfill & \text{if }m\ne 0.\hfill \end{array}$$

By (1.4), we have a distribution formula for $(h,q)$-Genocchi polynomials, which is shown by [3]

$${\tilde{G}}_{n+1,q}^{(\alpha ,h)}(x)=\frac{{[2]}_q}{{[2]}_{q^a}}{[a]}_{q^{\alpha }}^n\sum _{j=0}^{a-1}{(-1)}^j{q}^{jh}{\tilde{G}}_{n+1,q^a}^{(\alpha ,h)}\left(\frac{x+j}{a}\right).$$

By applying some elementary methods, we will give symmetric properties of weighted $(h,q)$-Genocchi polynomials and a weighted $(h,q)$-zeta-type function. Consequently, our applications seem to be interesting and worthwhile for further works of many mathematicians in analytic numbers theory.

2 On the $(h,q)$-zeta-type function

In this part, we firstly recall the $(h,q)$-zeta-type function with weight α which is derived in [3] as follows:

$${\tilde{\zeta }}_q^{(\alpha ,h)}(s,x)={[2]}_q\sum _{m=0}^{\mathrm{\infty }}\frac{{(-1)}^m{q}^{mh}}{{[m+x]}_{q^{\alpha }}^s},$$

(2.1)

where $q\in \mathbb{C}$, $h\in \mathbb{N}$ and $\mathrm{\Re }(s)>1$. It is clear that the special case $h=0$ and $q\to 1$ in (2.1) reduces to the ordinary Hurwitz-Euler zeta function. Now, we consider (2.1) in the following form:

$${\tilde{\zeta }}_{q^a}^{(\alpha ,h)}(s,bx+\frac{bj}{a})={[2]}_{q^a}\sum _{m=0}^{\mathrm{\infty }}\frac{{(-1)}^m{q}^{mah}}{{[m+bx+\frac{bj}{a}]}_{q^{a\alpha }}^s}.$$

By applying some basic operations to the above identity, that is, for any positive integers m and b, there exist unique non-negative integers k and i such that $m=bk+i$ with $0\le i\le b-1$. For $a\equiv 1(mod2)$ and $b\equiv 1(mod2)$. Thus, we can compute as follows:

(2.2)

From this, we can easily discover the following:

$$\begin{array}{c}\sum _{j=0}^{a-1}{(-1)}^j{q}^{jbh}{\tilde{\zeta }}_{q^a}^{(\alpha ,h)}(s,bx+\frac{bj}{a})\hfill \\ ={[a]}_{q^{\alpha }}^s{[2]}_{q^a}\sum _{j=0}^{a-1}{(-1)}^j{q}^{jbh}\sum _{i=0}^{b-1}{(-1)}^i{q}^{iah}\sum _{m=0}^{\mathrm{\infty }}\frac{{(-1)}^m{q}^{mbah}}{{[ab(m+x)+ai+bj]}_{q^{\alpha }}^s}.\hfill \end{array}$$

(2.3)

Replacing a by b and j by i in (2.2), we have the following:

$${\tilde{\zeta }}_{q^b}^{(\alpha ,h)}(s,ax+\frac{ai}{b})={[b]}_{q^{\alpha }}^s{[2]}_{q^b}\sum _{j=0}^{a-1}{(-1)}^j{q}^{jbh}\sum _{m=0}^{\mathrm{\infty }}\frac{{(-1)}^m{q}^{mbah}}{{[ab(m+x)+ai+bj]}_{q^{\alpha }}^s}.$$

By considering the above identity in (2.3), we can easily state the following theorem.

Theorem 1 The following identity is true:

$$\frac{{[2]}_{q^b}}{{[a]}_{q^{\alpha }}^s}\sum _{i=0}^{a-1}{(-1)}^i{q}^{ibh}{\tilde{\zeta }}_{q^a}^{(\alpha ,h)}(s,bx+\frac{bi}{a})=\frac{{[2]}_{q^a}}{{[b]}_{q^{\alpha }}^s}\sum _{i=0}^{b-1}{(-1)}^i{q}^{iah}{\tilde{\zeta }}_{q^b}^{(\alpha ,h)}(s,ax+\frac{ai}{b}).$$

Now, setting $b=1$ in Theorem 1, we have the following distribution formula:

$${\tilde{\zeta }}_q^{(\alpha ,h)}(s,ax)=\frac{{[2]}_q}{{[2]}_{q^a}{[a]}_{q^{\alpha }}^s}\sum _{i=0}^{a-1}{(-1)}^i{q}^{ih}{\tilde{\zeta }}_{q^a}^{(\alpha ,h)}(s,x+\frac{i}{a}).$$

(2.4)

Putting $a=2$ in (2.4) leads to the following corollary.

Corollary 1 The following identity holds true:

$${\tilde{\zeta }}_q^{(\alpha ,h)}(s,2x)=\frac{{[2]}_q}{{[2]}_{q^2}{[2]}_{q^{\alpha }}^s}({\tilde{\zeta }}_{q^2}^{(\alpha ,h)}(s,x)-q^h{\tilde{\zeta }}_{q^2}^{(\alpha ,h)}(s,x+\frac{1}{2})).$$

Taking $s=-m$ into Theorem 1, we have the symmetric property of $(h,q)$-Genocchi polynomials by the following theorem.

Theorem 2 The following identity is true:

$${[2]}_{q^b}{[a]}_{q^{\alpha }}^{m-1}\sum _{j=0}^{a-1}{(-1)}^i{q}^{ibh}{\tilde{G}}_{m,q^a}^{(\alpha ,h)}(bx+\frac{bi}{a})={[2]}_{q^a}{[b]}_{q^{\alpha }}^{m-1}\sum _{i=0}^{b-1}{(-1)}^i{q}^{iah}{\tilde{G}}_{m,q^b}^{(\alpha ,h)}(ax+\frac{ai}{b}).$$

Now also, setting $b=1$ and replacing x by $\frac{x}{a}$ in the above theorem, we can rewrite the following $(h,q)$-Genocchi polynomials with weight α:

$${\tilde{G}}_{n,q}^{(\alpha ,h)}(x)=\frac{{[2]}_q}{{[2]}_{q^a}}{[a]}_{q^{\alpha }}^{n-1}\sum _{i=0}^{a-1}{(-1)}^i{q}^{ih}{\tilde{G}}_{n,q^a}^{(\alpha ,h)}\left(\frac{x+i}{a}\right)(2\nmid a).$$

Due to Araci et al. [3], we develop as follows:

$$\begin{array}{rcl}\sum _{n=0}^{\mathrm{\infty }}{\tilde{G}}_{n,q}^{(\alpha ,h)}(x+y)\frac{t^n}{n!}& =& {[2]}_{q}t\sum _{m=0}^{\mathrm{\infty }}{(-1)}^m{q}^{mh}{e}^{t{[x+y+m]}_{q^{\alpha }}}\\ =& {[2]}_{q}t\sum _{m=0}^{\mathrm{\infty }}{(-1)}^m{q}^{mh}{e}^{t{[y]}_{q^{\alpha }}}{e}^{(q^{\alpha y}t){[x+m]}_{q^{\alpha }}}\\ =& \left(\sum _{n=0}^{\mathrm{\infty }}{[y]}_{q^{\alpha }}^n\frac{t^n}{n!}\right)(\sum _{n=0}^{\mathrm{\infty }}{q}^{\alpha (n-1)y}{\tilde{G}}_{n,q}^{(\alpha ,h)}(x)\frac{t^n}{n!}).\end{array}$$

By using the Cauchy product, we see that

$$\sum _{n=0}^{\mathrm{\infty }}(\sum _{j=0}^n\left(\genfrac{}{}{0ex}{}{n}{j}\right)q^{\alpha (j-1)y}{\tilde{G}}_{j,q}^{(\alpha ,h)}(x){[y]}_{q^{\alpha }}^{n-j})\frac{t^n}{n!}.$$

Thus, by comparing the coefficients of $\frac{t^n}{n!}$, we state the following corollary.

Corollary 2 The following equality holds true:

$${\tilde{G}}_{n,q}^{(\alpha ,h)}(x+y)=\sum _{j=0}^n\left(\genfrac{}{}{0ex}{}{n}{j}\right)q^{\alpha (j-1)y}{\tilde{G}}_{j,q}^{(\alpha ,h)}(x){[y]}_{q^{\alpha }}^{n-j}.$$

(2.5)

By using Theorem 2 and (2.5), we readily derive the following symmetric relation after some applications.

Theorem 3 The following equality holds true:

$$\begin{array}{c}{[2]}_{q^b}\sum _{i=0}^m\left(\genfrac{}{}{0ex}{}{m}{i}\right){[a]}_{q^{\alpha }}^{i-1}{[b]}_{q^{\alpha }}^{m-i}{\tilde{G}}_{i,q^a}^{(\alpha ,h)}(bx){\tilde{S}}_{m-i:q^b,h+i-1}^{(\alpha )}(a)\hfill \\ ={[2]}_{q^a}\sum _{i=0}^m\left(\genfrac{}{}{0ex}{}{m}{i}\right){[b]}_{q^{\alpha }}^{i-1}{[a]}_{q^{\alpha }}^{m-i}{\tilde{G}}_{i,q^b}^{(\alpha ,h)}(ax){\tilde{S}}_{m-i:q^a,h+i-1}^{(\alpha )}(b),\hfill \end{array}$$

where ${\tilde{S}}_{m:q,i}^{(\alpha )}(a)={\sum }_{j=0}^{a-1}{(-1)}^j{q}^{ji}{[j]}_{q^{\alpha }}^m$.

When $q\to 1$ into Theorem 3, it leads to the following corollary.

Corollary 3 The following identity holds true:

$$\begin{array}{c}\sum _{i=0}^m\left(\genfrac{}{}{0ex}{}{m}{i}\right)a^{i-1}{b}^{m-i}{G}_i(bx)S_{m-i}(a)\hfill \\ =\sum _{i=0}^m\left(\genfrac{}{}{0ex}{}{m}{i}\right)b^{i-1}{a}^{m-i}{G}_i(ax)S_{m-i}(b),\hfill \end{array}$$

where $S_m(a)={\sum }_{j=0}^{a-1}{(-1)}^j{j}^m$ and $G_n(x)$ are called the ordinary Genocchi polynomials which are defined via the following generating function:

$$\sum _{n=0}^{\mathrm{\infty }}{G}_n(x)\frac{t^n}{n!}=\frac{2t}{e^t+1}{e}^{xt}.$$