Skip to main content

Advertisement

A note on the Von Staudt-Clausen’s theorem for the weighted q-Genocchi numbers

Article metrics

  • 1117 Accesses

  • 4 Citations

Abstract

Recently, the Von Staudt-Clausen theorem for q-Euler numbers was introduced by Kim (Russ. J. Math. Phys. 20(1):33-38, 2013) and Araci et al. have also studied this theorem for q-Genocchi numbers (see Araci et al. in Appl. Math. Comput. 247:780-785, 2014) based on the work of Kim et al. In this paper, we give the corresponding Von Staudt-Clausen theorem for the weighted q-Genocchi numbers and also prove the Kummer-type congruences for the generated weighted q-Genocchi numbers.

Introduction and preliminaries

As is well known, a theorem including the fractional part of Bernoulli numbers, which is called the Von Staudt-Clausen theorem, was introduced by Karl Von Staudt and Thomas Clausen (see [1]). In [2], Kim has studied the Von Staudt-Clausen theorem for the q-Euler numbers and Araci et al. have introduced the Von Staudt-Clausen theorem associated with q-Genocchi numbers.

Let p be a fixed odd prime number. Throughout this paper, \(\mathbb{Z}_{p}\), \(\mathbb{Q}_{p}\) and \(\mathbb{C}_{p}\) will denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of the algebraic closure \(\mathbb{Q}_{p}\). Let us assume that q is an indeterminate in \(\mathbb {C}_{p}\) with \(|1-q|_{p}< p^{-\frac{1}{1-p}}\) where \(|\cdot|_{p}\) is a p-adic norm. The q-extension of x is defined by \([x]_{q}=\frac{1-q^{x}}{1-q}\). Note that \(\lim_{q\rightarrow1}[x]_{q}=x\). For \(f\in C(\mathbb{Z}_{p})\) = the space of all continuous functions on \(\mathbb{Z}_{p}\), the fermionic p-adic q-integral on \(\mathbb{Z}_{p}\) is defined by Kim to be

$$\begin{aligned} \int_{\mathbb{Z}_{p}}f(x)\,d\mu_{-q} (x) = \lim _{N\rightarrow\infty} \frac{1}{[p^{N}]_{-q} }\sum_{x=0}^{p^{N}-1}f(x) (-q)^{x}\quad (\mbox{see [2--6]}). \end{aligned}$$
(1)

From (1), we note that

$$\begin{aligned} q \int_{\mathbb{Z}_{p}}f(x+1)\,d\mu_{-q} (x) + \int_{\mathbb{Z}_{p}} f(x)\,d\mu_{-q} (x)=[2]_{q} f(0). \end{aligned}$$
(2)

From \(n\in\mathbb{N}\), we have

$$ \begin{aligned}[b] &q^{n} \int_{\mathbb{Z}_{p}}f(x+n)\,d \mu_{-q} (x) + (-1)^{n-1} \int_{\mathbb {Z}_{p}} f(x)\,d \mu_{-q} (x) \\ &\quad=[2]_{q} \sum_{l=0}^{n-1} f(l) (-1)^{n-l-1}q^{l} \quad(\mbox{see [4]}). \end{aligned} $$
(3)

Let \(d\in\mathbb{N}\) with \(d \equiv1\ (\operatorname{mod}\ 2)\) and \((p,d)=1\). Then we set

$$\begin{aligned} x = x_{d} =\lim_{\overleftarrow{N}} \mathbb{Z}/dp^{N} \mathbb{Z},\qquad X^{*}= \bigcup_{0< a<dp,(a,p)=1} a+dp \mathbb{Z}_{p} \end{aligned}$$

and \(a+dp^{N} \mathbb{Z}_{p}=\{ x\in X |x\equiv a\ (\operatorname{mod}\ dp^{N})\}\) where \(a\in\mathbb{Z}\) lies in \(0\leq a < dp^{N}\). It is well known that

$$\begin{aligned} \int_{X} f(x)\,d\mu_{-q} (x) =\int _{\mathbb{Z}_{p}} f(x)\,d\mu_{-q} (x), \quad\mbox{where } f\in C( \mathbb{Z}_{p})\ (\mbox{see [2--6]}). \end{aligned}$$
(4)

Recently, the weighted q-Euler numbers were introduced by the generating function to be

$$\begin{aligned} \sum_{n=0}^{\infty}E_{n,q}^{(\alpha)} \frac{t^{n}}{n!}= \int_{\mathbb{Z}_{p}} e^{[x]_{q^{\alpha}} t}\,d\mu_{-q}(x) = \sum_{n=0}^{\infty}\biggl( \int _{\mathbb{Z}_{p}} [x]_{q^{\alpha}}^{n} d\mu _{-q}(x) \biggr) \frac{t^{n}}{n!} \quad(\mbox{see [5, 7]}). \end{aligned}$$
(5)

Thus, by (5), we get

$$\begin{aligned} E_{n,q}^{(\alpha)} (x) = \int_{\mathbb{Z}_{p}} [x]_{q^{\alpha}}^{n} \,d\mu_{-q}(x) \quad(\mbox{see [5, 8]}), \end{aligned}$$

where \(\alpha\in\mathbb{C}_{p}\). Many researchers have studied the weighted q-Euler numbers and q-Genocchi numbers in the recent decade (see [116]).

From (5), Araci defined the weighted q-Genocchi numbers as follows:

$$\begin{aligned} \sum_{n=0}^{\infty}G_{n,q}^{(\alpha)} \frac{t^{n}}{n!}= t \int_{\mathbb{Z}_{p}} e^{[x]_{q^{\alpha}} t}\,d\mu_{-q}(x) = \sum_{n=0}^{\infty}\biggl( \int _{\mathbb{Z}_{p}} [x]_{q^{\alpha}}^{n} \,d\mu _{-q}(x) \biggr) \frac{t^{n+1}}{n!}. \end{aligned}$$
(6)

By (6), we get

$$\begin{aligned} \frac{G_{n+1,q}^{(\alpha)} }{n+1} = \int_{\mathbb{Z}_{p}} [x]_{q^{\alpha}}^{n} \,d\mu_{-q}(x),\qquad G_{0,q}^{(\alpha)}=0. \end{aligned}$$
(7)

The weighted q-Genocchi polynomials are also defined by

$$\begin{aligned} \sum_{n=0}^{\infty}G_{n,q}^{(\alpha)} (x) \frac{t^{n}}{n!}= t \int _{\mathbb{Z}_{p}} e^{[x+y]_{q^{\alpha}} t}\,d\mu_{-q}(x). \end{aligned}$$
(8)

Thus, by (8), we have

$$\begin{aligned} \frac{G_{n+1,q}^{(\alpha)} (x)}{n+1} = \int_{\mathbb{Z}_{p}} [x+y]_{q^{\alpha}}^{n} \,d\mu_{-q}(y)\quad (n\geq0). \end{aligned}$$
(9)

Let us assume that χ is a Dirichlet character with conductor \(d\in \mathbb{N}\) with \(d\equiv1\ (\operatorname{mod}\ 2)\). Then we defined the generalized weighted q-Genocchi numbers attached to χ as follows:

$$\begin{aligned} \frac{G_{n+1,q,\chi}^{(\alpha)} }{n+1} = \int_{X} \chi(x)[x]_{q^{\alpha}}^{n} \,d\mu_{-q}(x). \end{aligned}$$
(10)

From (10), we have

$$\begin{aligned} \frac{G_{n+1,q,\chi}^{(\alpha)}}{n+1} =& \int_{X} \chi(x)[x]_{q^{\alpha}}^{n} \,d\mu_{-q}(x) \\ =& \lim_{N\rightarrow\infty} \frac{1}{[dp^{N}]_{-q}} \sum _{x=0}^{dp^{N}-1} \chi(x) (-1)^{x} [x]_{q^{\alpha}}^{n} \\ =& \frac{[d]_{q^{\alpha}}^{n}}{[d]_{-q}}\sum_{k=0}^{d-1} (-1)^{k} \chi(k)q^{k} \Biggl( \lim_{N\rightarrow\infty} \frac{1}{[p^{N}]_{-q^{d}}} \sum_{x=0}^{p^{N}-1} \biggl[x+ \frac{k}{d} \biggr]_{q^{d\alpha}}(-1)^{x} q^{dx} \Biggr) \\ =& \frac{[d]_{q^{\alpha}}^{n}}{[d]_{-q}}\sum_{k=0}^{d-1} (-1)^{k} \chi(k)q^{k} \frac{G_{n+1,q^{d}}^{(\alpha)} (\frac{k}{d} )}{n+1}. \end{aligned}$$
(11)

Theorem 1.1

Let χ be the Dirichlet character with conductor \(d \in\mathbb{N}\) with \(d\equiv1\ (\operatorname{mod}\ 2)\). For \(n \in\mathbb{N}^{*}=\mathbb{N}\cup\{0\}\), we have

$$\begin{aligned} G_{n,q,\chi}^{(\alpha)} = \frac{[d]_{q^{\alpha}}^{n}}{[d]_{-q}}\sum _{k=0}^{d-1} (-1)^{k} \chi(k)q^{k} G_{n,q^{d}}^{(\alpha)} \biggl(\frac{k}{d} \biggr). \end{aligned}$$

Next we give a familiar theorem, which is known as the Von Staudt-Clausen theorem.

Lemma 1.2

(Von Staudt-Clausen theorem)

Let n be an even and positive integer. Then

$$\begin{aligned} B_{n} + \sum_{p-1|n, p:\mathrm{prime}} \frac{1}{p} \in \mathbb{Z}. \end{aligned}$$

Notice that \(pB_{n}\) is a p-adic integer where p is an arbitrary prime number, n is an arbitrary integer and also \(B_{n}\) is a Bernoulli number as in [1]. The purpose of this paper is to show that the weighted q-Genocchi numbers can be described by a Von Staudt-Clausen-type theorem. Finally, we prove a Kummer-type congruence for the generated weighted q-Genocchi numbers.

Von Staudt-Clausen theorems

From (10), we have

$$\begin{aligned} \frac{G_{n+1,q}^{(\alpha)}}{n+1} =\int_{\mathbb{Z}_{p}} [x]_{q^{\alpha}}^{n} \,d\mu_{-q}(x) = \frac{[2]_{q}}{2} \int_{\mathbb{Z}_{p}} q^{x} [x]_{q^{\alpha}}^{n} d \mu_{-1}(x). \end{aligned}$$
(12)

Thus, by (12), we have

$$\begin{aligned} \lim_{q\rightarrow1}\frac{G_{n+1,q}^{(\alpha)}}{n+1} =\frac{G_{n+1}}{n+1}=\int _{\mathbb{Z}_{p}}x^{n} \,d\mu_{-1}(x) \quad(\mbox{see [2--6, 15]}). \end{aligned}$$

In [2], Kim introduced the following inequality:

$$\begin{aligned} \Biggl\vert \sum_{j=0}^{p-1} (-1)^{j} [j]_{q^{\alpha}}q^{j} \Biggr\vert \leq1. \end{aligned}$$
(13)

Let us define the following equality: for \(k\geq1\),

$$\begin{aligned} L_{n-1}^{(\alpha)} (k)=[0]_{q^{\alpha}}^{n-1}- q[1]_{q^{\alpha}}^{n-1} +\cdots+ \bigl[p^{k}-1 \bigr]_{q^{\alpha}}^{n-1}q^{p^{k}-1}. \end{aligned}$$
(14)

From (3), we note that

$$\begin{aligned} q^{d}\frac{G_{n+1,q^{d}}^{(\alpha)}(d)}{n+1} +\frac{G_{n+1,q^{d}}^{(\alpha)}}{n+1} = [2]_{q} \sum_{l=0}^{d-1} [l]_{q^{d}}^{n} (-1)^{l} q^{l} , \end{aligned}$$
(15)

where \(d\in\mathbb{N}\) with \(d\equiv1\ (\operatorname{mod}\ 2)\). By (14) and (12), we get

$$\begin{aligned} \lim_{k\rightarrow\infty} nL_{n-1}^{(\alpha)}(k)= \frac {2}{[2]_{q}}G_{n,q}^{(\alpha)}. \end{aligned}$$

By (14), we get

$$\begin{aligned} & L_{n-1}^{(\alpha)} (k+1) \\ &\quad= \sum_{a=0}^{p^{k+1}-1} (-1)^{a} q^{a} [a]_{q^{\alpha}}^{n-1} \\ &\quad= \sum_{a=0}^{p^{k}-1} \sum _{j=0}^{p-1} (-1)^{a+jp^{k}}q^{a+jp^{k}} \bigl[a+jp^{k} \bigr]_{q^{\alpha}}^{n-1} \\ &\quad= \sum_{a=0}^{p^{k}-1}\sum _{j=0}^{p-1} (-1)^{a+jp^{k}}q^{a+jp^{k}} \bigl([a]_{q^{\alpha}}+q^{\alpha a} \bigl[jp^{k} \bigr]_{q^{\alpha}} \bigr)^{n-1} \\ &\quad= \sum_{a=0}^{p^{k}-1}\sum _{j=0}^{p-1} \sum_{l=0}^{n-1} \binom{n-1}{l} [a]_{q^{\alpha}}^{n-1-l}(-1)^{a+j} q^{a \alpha l} \bigl[jp^{k} \bigr]_{q^{\alpha}}^{l} q^{a+jpk} \\ &\quad= \sum_{a=0}^{p^{k}-1}\sum _{j=0}^{p-1}\sum_{l=0}^{n-1} \binom{n-1}{l} [a]_{q^{\alpha}}^{n-1-l}(-1)^{a+j} q^{a(\alpha l+1)+jp^{k}} \bigl[p^{k} \bigr]_{q^{\alpha}}^{l} [j]_{q^{\alpha}p^{k}}^{l} \\ &\quad= \sum_{a=0}^{p^{k}-1}(-1)^{a} q^{a} [a]_{q^{\alpha}}^{n-1} \frac{[2]_{q^{ p^{2k}}}}{[2]_{q^{p^{k}}}} \\ &\qquad{}+ \sum _{a=0}^{p^{k}-1}\sum_{j=0}^{p-1} \sum_{l=1}^{n-1}\binom{n-1}{l} [a]_{q^{\alpha}}^{n-1-l}(-1)^{a+j} q^{a(\alpha l+1)+jp^{k}} \bigl[p^{k} \bigr]_{q^{\alpha}}^{l} [j]_{q^{\alpha p^{k}}}^{l} \\ &\quad= \sum_{a=0}^{p^{k}-1}\sum _{j=0}^{p-1}\sum_{l=0}^{n-1} \binom{n-1}{l} [a]_{q^{\alpha}}^{n-1-l}(-1)^{a+j} q^{a(\alpha+l)+jp^{k}} \bigl[p^{k} \bigr]_{q^{\alpha}}^{l} [j]_{q^{\alpha}p^{k}}^{l} \\ &\quad= \sum_{a=0}^{p^{k}-1}(-1)^{a} q^{a} [a]_{q^{\alpha}}^{n-1} \frac {[2]_{q^{p^{2k}}}}{[2]_{q^{p^{k}}}} \\ &\qquad{}+\sum_{a=0}^{p^{k}-1}\sum _{j=0}^{p-1} \sum_{l=0}^{n-1} \binom{n-1}{l} [a]_{q^{\alpha}}^{n-1-l}(-1)^{a+j} q^{a(\alpha l+1)+jp^{k}} \bigl[p^{k} \bigr]_{q^{\alpha}}^{l} [j]_{q^{\alpha p^{k}}}^{l}. \end{aligned}$$
(16)

Thus, by (16), we get

$$\begin{aligned} L_{n-1}^{(\alpha)} (k+1)\equiv\sum _{a=0}^{p^{k}-1} [a]_{q^{\alpha}}^{n-1}(-1)^{a}q^{a} \ \bigl(\operatorname{mod}\ \bigl[p^{k} \bigr]_{q^{\alpha}} \bigr). \end{aligned}$$
(17)

From (16), we have

$$\begin{aligned} &\sum_{a=0}^{p^{k+1}-1} (-1)^{a} [a]_{q^{\alpha}}^{n-1}q^{a} \\ &\quad= \sum_{a=0}^{p-1}\sum _{j=0}^{p^{k}-1} (-1)^{a+pj}[a+pj]_{q^{\alpha}}^{n-1}q^{a+pj} \\ &\quad= \sum_{a=0}^{p-1}(-1)^{a}q^{a} \sum_{j=0}^{p^{k}-1} (-1)^{j} q^{pj} \bigl([a]_{q^{\alpha}}+ q^{\alpha a}[p]_{q^{\alpha}} [j]_{q^{\alpha p}} \bigr)^{n-1} \\ &\quad= \sum_{a=0}^{p-1}\sum _{j=0}^{p^{k}-1} \sum_{l=0}^{n-1} \binom{n-1}{l} (-1)^{a+j} q^{a+pj} [a]_{q^{\alpha}}^{n-1-l} q^{\alpha a l} [p]_{q^{\alpha}}^{l} [j]_{q^{p\alpha}}^{l} \\ &\quad= \sum_{a=0}^{p-1}(-1)^{a} q^{a} [a]_{q^{\alpha}}^{n-1} \frac{[2]_{q^{ p^{k+1}}}}{[2]_{q^{p}}} \\ &\qquad{} + \sum_{a=0}^{p-1}\sum _{j=0}^{p^{k}-1} \sum_{l=1}^{n-1} \binom{n-1}{l} (-1)^{a+j}q^{a+pj+\alpha al} [a]_{q^{\alpha}}^{n-1-l} [p]_{q^{\alpha}}^{l} [j]_{q^{p\alpha}}^{l} \\ &\quad= \sum_{a=0}^{p-1}(-1)^{a} q^{a} [a]_{q^{\alpha}}^{n-1} \ \bigl(\operatorname{mod}\ [p]_{q^{\alpha}} \bigr). \end{aligned}$$
(18)

Therefore, by (17) and (18), we obtain the following theorem.

Theorem 2.1

Let \(L_{n-1}^{(\alpha)} (k)= \sum_{a=0}^{p^{k}-1} (-1)^{a} [a]_{q^{\alpha}}^{n-1}\). Then we have

$$\begin{aligned} L_{n-1}^{(\alpha)} (k+1)= \sum_{a=0}^{p^{k}-1} [a]_{q^{\alpha}}^{n-1}(-1)^{a} q^{a}. \end{aligned}$$

Furthermore

$$\begin{aligned} \sum_{a=0}^{p^{k}-1} [a]_{q^{a}}^{n-1}(-1)^{a} q^{a} \alpha \ \bigl(\operatorname{mod}\ \bigl[p^{k} \bigr]_{q^{\alpha}} \bigr) \equiv\sum_{a=0}^{p-1}(-1)^{a} q^{a}[a]_{q^{\alpha}}^{n-1} \ \bigl(\operatorname{mod}\ [p]_{q^{\alpha}} \bigr). \end{aligned}$$

By Theorem 2.1, we get

$$\begin{aligned} \sum_{a=0}^{p-1} (-1)^{a} n [a]_{q^{\alpha}}^{n-1} q^{a} =\int _{X} [x]_{q^{\alpha}}^{n-1}\,d \mu_{-q}(x) \equiv G_{n,q}^{(\alpha)} \ \bigl(\operatorname{mod}\ [p]_{q} \bigr). \end{aligned}$$
(19)

Therefore, by (19), we have the following theorem.

Theorem 2.2

For \(n\geq1\), we have

$$\begin{aligned} \sum_{a=0}^{p-1} (-1)^{a} n [a]_{q^{\alpha}}^{n-1}=G_{n,q}^{(\alpha)} \ \bigl( \operatorname{mod}\ [p]_{q} \bigr). \end{aligned}$$

From (17) and (19), we note that

$$\begin{aligned} G_{n+1,q}^{(\alpha)} + n \sum_{a=0}^{p-1} (-1)^{a+1} [a]_{q^{\alpha}}^{n-1}q^{a} \in \mathbb{Z}_{p} \quad(n\geq1). \end{aligned}$$

Corollary 2.3

For \(n\geq1\), we have

$$\begin{aligned} G_{n+1,q}^{(\alpha)} + n \sum_{a=0}^{p-1} (-1)^{a+1} [a]_{q^{\alpha}}^{n-1}q^{a} \in \mathbb{Z}_{p}. \end{aligned}$$

Let \(n\geq1\). Then we observe that

$$\begin{aligned} \biggl\vert \frac{G_{n+1,q}^{(\alpha)}}{n+1} \biggr\vert _{p} &= \Biggl\vert \frac{G_{n+1,q}^{(\alpha)}}{n+1} -\sum_{a=0}^{p-1}(-1)^{a}[a]_{q^{\alpha}}^{n} q^{a} + \sum_{a=0}^{p-1}(-1)^{a}q^{a} [a]_{q^{\alpha}}^{n} \Biggr\vert _{p} \\ &\leq \max \Biggl\{ \Biggl\vert \frac{G_{n+1,q}^{(\alpha)}}{n+1} -\sum _{a=0}^{p-1}(-1)^{a}[a]_{q^{\alpha}}^{n} \Biggr\vert _{p}, \Biggl\vert \sum_{a=0}^{p-1}(-1)^{a}q^{a} [a]_{q^{\alpha}}^{n} \Biggr\vert _{p} \Biggr\} \leq1. \end{aligned}$$
(20)

Therefore, we obtain the following theorem.

Theorem 2.4

For \(n\geq1\), we have

$$\begin{aligned} \frac{G_{n+1,q}^{(\alpha)}}{n+1} \in\mathbb{Z}_{p}. \end{aligned}$$

Let χ be the Dirichlet character \(d\in\mathbb{N}\) with \(d\equiv1\ (\operatorname{mod}\ 2)\). The generalized weighted q-Genocchi numbers attached to χ are introduced as follows:

$$\begin{aligned} \sum_{n=0}^{\infty}G_{n,q,\chi}^{(\alpha)} \frac{t^{n}}{n!} =& [2]_{q} t \sum _{m=0}^{\infty}(-1)^{m} \chi(m)e^{[m]_{q^{\alpha}} t} \\ =& t \int_{X} \chi(x) e^{[x]_{q^{\alpha}}t}\,d \mu_{-q}(x). \end{aligned}$$
(21)

Let \(\overline{f}=[f,p]\) be the least common multiple of the conductor f of χ and p. By (21), we get

$$\begin{aligned} G_{n,q,\chi}^{(\alpha)} = n\int_{X} \chi(x) [x]_{q^{\alpha}}^{n-1}\,d\mu _{-q}(x) = n \lim_{n\rightarrow\infty} \sum_{x=0}^{fp^{N}-1} \chi(x) (-1)^{x} [x]_{q^{\alpha}}^{n-1}. \end{aligned}$$
(22)

Thus, we have

$$\begin{aligned} G_{n,q,\chi}^{(\alpha)} =& n \lim_{\rho\rightarrow\infty} \sum_{1\leq a\leq\overline {f}p^{\rho}, (a,p)=1} \chi(a) (-1)^{a} q^{a} [a]_{q^{\alpha}}^{n-1} \\ &{}+ n[p]_{q^{\alpha}}^{n-1} \chi(p) \lim_{\rho\rightarrow\infty} \sum_{1\leq a\leq\overline{f}p^{\rho}, (a,p)=1}^{\overline{f}p^{\rho}-1} \chi(a) (-1)^{a}q^{ap}[a]_{q^{\alpha}p}^{n-1} \\ =& n \lim_{\rho\rightarrow\infty} \sum_{1\leq a\leq\overline{f}p^{p}, (a,p)=1} \chi(a) (-1)^{a} q^{a} [a]_{q^{\alpha}}^{n-1} +a[p]_{q^{\alpha}}^{n-1}\chi (p)G_{n,q^{p},\chi}^{(\alpha)}. \end{aligned}$$
(23)

Therefore, by (23), we obtain the following theorem.

Theorem 2.5

For \(n\geq1\), we have

$$\begin{aligned} n \lim_{\rho\rightarrow\infty} \sum _{1\leq a\leq\overline{f}p^{\rho}, (a,p)=1} \chi(a) (-1)^{a}q^{a}[a]_{q^{\alpha}}^{n-1} = G_{n,q,\chi}^{(\alpha)}-[p]_{q^{\alpha}}^{n-1} \chi(p)G_{n,q^{p}, \chi }^{(\alpha)}. \end{aligned}$$
(24)

Assume that w is the Teichmüller character by modp. For \(a\in X^{*}\), set \(\langle a\rangle_{\alpha}=\langle a:q\rangle_{\alpha}=\frac{[a]_{q^{\alpha}}}{w(a)}\). Note that \(|\langle a\rangle_{\alpha}-1|_{p}< p^{\frac{1}{p-1}}\), where \(\langle a\rangle^{s}=\exp (s \log \langle a\rangle)\) for \(s\in\mathbb{Z}_{p}\). For \(s\in\mathbb{Z}_{p}\), we define the weighted p-adic l-function associated with \(G_{n,q,\chi}^{(\alpha)}\) as follows:

$$\begin{aligned} l_{p,q}^{(\alpha)}(s, \chi)= \lim_{\rho\rightarrow\infty} \sum _{1\leq a\leq\overline{f}p^{\rho}, (a,p)=1} \chi(a) (-1)^{a} \langle a \rangle_{\alpha}^{-s}q^{a}= \int_{X^{*}} \chi(x)\langle x\rangle_{\alpha}^{-s}\,d\mu_{-q}(x). \end{aligned}$$

For \(k\geq1\),

$$\begin{aligned} & k l_{p,q} \bigl(1-k,\chi w^{k-1} \bigr) \\ &\quad= k \lim_{\rho\rightarrow\infty} \sum_{1\leq a\leq\overline{f}p^{\rho}} \chi(a) (-1)^{a}q^{a} [a]_{q^{\alpha}}^{k-1} \\ &\quad= k \int_{X} \chi(x)[x]_{q^{\alpha}}^{k-1} \,d\mu_{-q}(x) - k\int_{p X} \chi(x)[x]_{q^{\alpha}}^{k-1} \,d\mu_{-q}(x) \\ &\quad= k\int_{X} \chi(x)[x]_{q^{\alpha}}^{k-1} \,d\mu_{-q}(x)- \frac{k[2]_{q}\chi(p)}{[2]_{q^{p}}} [p]_{q^{\alpha}}^{k-1} \int_{X} \chi(x) [x]_{q^{\alpha p}}^{k-1}\,d \mu_{-q^{p}}(x) \\ &\quad= G_{x,q,\chi}^{(\alpha)} - \frac{[2]_{q}}{[2]_{q^{p}}}\chi (p)[p]_{q^{\alpha}}^{k-1} G_{k,q^{p},\chi}^{(\alpha)}. \end{aligned}$$

It is easy to show that

$$\begin{aligned}[b] \langle a\rangle_{\alpha}^{p^{n}} &= \exp \bigl(p^{n} \log \langle a\rangle_{\alpha}\bigr) \\ &= 1+ p^{n} \log\langle a\rangle_{\alpha}+ \frac{(p^{n} \log_{p}\langle a\rangle_{\alpha})^{2}}{2!}+ \cdots \\ &\equiv1 \ \bigl(\operatorname{mod}\ p^{n} \bigr). \end{aligned} $$

So, by the definition of \(l_{p,q}^{(\alpha)}(1-k,x)\), we get

$$\begin{aligned}[b] l_{p,q}^{(\alpha)}(-k,\chi) &= \lim_{\rho\rightarrow\infty} \sum _{1\leq a\leq\overline{f}p^{\rho}, (a,p)=1} \chi(a) (-1)^{a} q^{a} \langle a\rangle_{\alpha}^{k} \\ &\equiv \lim_{\rho\rightarrow\infty} \sum_{1\leq a\leq\overline {f}p^{\rho}, (a,p)=1} \chi(a) (-1)^{a} q^{a} \langle a\rangle_{\alpha}^{k'} \ \bigl(\operatorname{mod}\ p^{n} \bigr), \end{aligned} $$

where \(k\equiv k' \ (\operatorname{mod}\ p^{n} (p-1))\). Namely, we have

$$\begin{aligned} l_{p,q}^{(\alpha)} \bigl(-k,\chi w^{k} \bigr)\equiv l_{p,q}^{(\alpha)} \bigl(-k',\chi w^{k'} \bigr) \ \bigl(\operatorname{mod}\ p^{n} \bigr). \end{aligned}$$

Theorem 2.6

For \(k\equiv k'\ (\operatorname{mod}\ p^{n} (p-1))\), we have

$$\begin{aligned} \frac{G_{k+1,q,\chi}^{(\alpha)}}{k+1}-\frac{[2]_{q}}{[2]_{q^{p}}}\frac {G_{k+1,q^{p},\chi}^{(\alpha)}}{k+1} \equiv \frac{G_{k'+1,q,\chi}^{(\alpha)}}{k'+1}-\frac{[2]_{q}}{[2]_{q^{p}}}\frac {G_{k'+1,q^{p},\chi}^{(\alpha)}}{k'+1} \ \bigl(\operatorname{mod}\ p^{n} \bigr). \end{aligned}$$

References

  1. 1.

    Araci, S, Acikgoz, M, Sen, E: On the von Staudt-Clausen’s theorem associated with q-Genocchi numbers. Appl. Math. Comput. 247, 780-785 (2014)

  2. 2.

    Kim, T: On the von Staudt-Clausen theorem for q-Euler numbers. Russ. J. Math. Phys. 20(1), 33-38 (2013)

  3. 3.

    Kim, T, Jang, L-C, Kim, Y-H: Some properties on the p-adic invariant integral on \(\mathbb {Z}_{p}\) associated with Genocchi and Bernoulli polynomials. J. Comput. Anal. Appl. 13(7), 1201-1207 (2011)

  4. 4.

    Kim, T: New approach to q-Euler polynomials of higher order. Russ. J. Math. Phys. 17(2), 218-225 (2010)

  5. 5.

    Kim, T: Identities on the weighted q-Euler numbers and q-Bernstein polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 22(1), 7-12 (2012)

  6. 6.

    Kim, T, Kim, Y-H, Ryoo, CS: Some identities on the weighted q-Euler numbers and q-Bernstein polynomials. J. Inequal. Appl. 2011, 64 (2011)

  7. 7.

    Park, J-W: New approach to q-Bernoulli polynomials with weight or weak weight. Adv. Stud. Contemp. Math. 24(1), 39-44 (2014)

  8. 8.

    Ryoo, CS: A note on the weighted q-Euler numbers and polynomials. Adv. Stud. Contemp. Math. 21(1), 47-54 (2011)

  9. 9.

    Araci, S, Acikgoz, M, Jolany, H, Seo, J: A unified generating function of the q-Genocchi polynomials with their interpolation functions. Proc. Jangjeon Math. Soc. 15(2), 227-233 (2012)

  10. 10.

    Bayad, A, Kim, T: Identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials. Adv. Stud. Contemp. Math. 20(2), 247-253 (2010)

  11. 11.

    Gaboury, S, Tremblay, R, Fugere, B-J: Some explicit formulas for certain new classes of Bernoulli, Euler and Genocchi polynomials. Proc. Jangjeon Math. Soc. 17(1), 115-123 (2014)

  12. 12.

    Jang, LC, Bell, ET: A study on the distribution of twisted q-Genocchi polynomials. Adv. Stud. Contemp. Math. 18(2), 181-189 (2009)

  13. 13.

    Jeong, J-H, Park, J-W, Rim, S-H: New approach to the analogue of Lebesgue-Radon-Nikodym theorem with respect to weighted p-adic q-measure on \(\mathbb{Z}_{p}\). J. Comput. Anal. Appl. 15(7), 1310-1316 (2013)

  14. 14.

    Kim, DS, Kim, T: q-Bernoulli polynomials and q-umbral calculus. Sci. China Math. 57(9), 1867-1874 (2014)

  15. 15.

    Kim, T, Kim, DS, Dolgy, DV, Rim, SH: Some identities on the Euler numbers arising from Euler basis polynomials. Ars Comb. 109, 433-446 (2013)

  16. 16.

    Pak, HK, Rim, SH, Jeong, J: A note on the analogue of Lebesgue-Radon-Nikodym theorem with respect to weighted p-adic q-measure on \(\mathbb{Z}_{p}\). J. Inequal. Appl. 2013, 15 (2013)

Download references

Acknowledgements

This paper was supported by Konkuk University in 2015.

Author information

Correspondence to Lee-Chae Jang.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to this work. All authors read and approved the final manuscript.

Rights and permissions

Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

MSC

  • 11B68
  • 11S40

Keywords

  • Genocchi number
  • weighted q-Genocchi number
  • weighted q-Euler number
  • Von Staudt-Clausen theorem