1 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 [1–16]).

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.

2 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

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