The modified degenerate q-Bernoulli polynomials arising from p-adic invariant integral on $\mathbb{Z}_{p}$

Abstract

Dolgy et al. introduced the modified degenerate Bernoulli polynomials, which are different from Carlitz’s degenerate Bernoulli polynomials (see Dolgy et al. in Adv. Stud. Contemp. Math. (Kyungshang) 26(1):1-9, 2016). In this paper, we study some explicit identities and properties for the modified degenerate q-Bernoulli polynomials arising from the p-adic invariant integral on $\mathbb{Z}_{p}$.

1 Introduction

For a fixed prime number p, ${\mathbb{Z}}_{p}$ refers to the ring of p-adic integers, ${\mathbb{Q}}_{p}$ to the field of p-adic rational numbers, and ${\mathbb{C}}_{p}$ to the completion of algebraic closure of ${\mathbb{Q}}_{p}$. The p-adic norm $\vert \cdot \vert _{p}$ is normalized as $\vert p\vert _{p}=\frac{1}{p}$. Let q be in ${\mathbb{C}}_{p}$ with $\vert q-1\vert _{p}< p^{-\frac{1}{p-1}}$ and $q^{x} = \exp(x \log q)$ for $\vert x\vert _{p} < 1$. Then the q-analogue of x is defined to be $[x]_{q}=\frac{1-q^{x}}{1-q}$.

The Bernoulli polynomials are given by the generating function

$$ \biggl( \frac{t}{e^{t}-1} \biggr) e^{xt}=\sum _{n=0} ^{\infty }B_{n}(x)\frac{t ^{n}}{n!} \quad \bigl(\mbox{see [1--25]}\bigr). $$

(1.1)

When $x=0$, $B_{n}=B_{n}(0)$ are called Bernoulli numbers.

Carlitz [4, 5, 8] defined the degenerate Bernoulli polynomials as follows:

$$ \frac{t}{(1+\lambda t)^{\frac{1}{\lambda}} -1} (1+\lambda t)^{\frac{x}{ \lambda}}=\sum _{n=0} ^{\infty}\beta_{n}(x\vert \lambda) \frac{t^{n}}{n!}. $$

(1.2)

When $x=0$, $\beta_{n}(0\vert \lambda)=\beta_{n}(\lambda)$ are called Carlitz’s degenerate Bernoulli numbers.

From (1.2) we note that

$$\begin{aligned} \sum_{n=0} ^{\infty} \lim_{\lambda\rightarrow0} \beta _{n}(x\vert \lambda)\frac{t^{n}}{n!} =& \lim_{\lambda\rightarrow0} \frac{t}{(1+\lambda t)^{\frac{1}{\lambda}} -1} (1+ \lambda t)^{\frac{x}{\lambda}} \\ =& \biggl( \frac{t}{e^{t}-1} \biggr) e^{xt} \\ =& \sum_{n=0} ^{\infty}B_{n}(x) \frac{t^{n}}{n!}. \end{aligned}$$

(1.3)

Using the derivation given in (1.3), we have

$$ \lim_{\lambda\rightarrow0} \beta_{n}(x\vert \lambda) = B_{n}(x)\quad ( n \geq0). $$

(1.4)

Let $f(x)$ be a uniformly differentiable function on ${\mathbb{Z}} _{p}$. Then the p-adic invariant integral on ${\mathbb{Z}}_{p}$ (also called the Volkenborn integral on ${\mathbb{Z}}_{p}$) is defined by

$$ \int_{\mathbb{Z}_{p}}f(x)\,d\mu_{0}(x)= \lim _{N\rightarrow\infty}\frac{1}{p ^{N}}\sum_{n=0} ^{p^{N}-1}f(x)\quad \bigl(\mbox{see [1, 9, 10, 15, 17]}\bigr). $$

(1.5)

By using the formula defined in (1.1) we note that

$$ \int_{\mathbb{Z}_{p}}f_{1}(x)\,du_{0}(x) - \int_{\mathbb{Z}_{p}}f(x)\,du _{0}(x)= f'(0) $$

(1.6)

and

$$ \int_{\mathbb{Z}_{p}}f_{n}(x)\,du_{0}(x) - \int_{\mathbb{Z}_{p}}f(x)\,du _{0}(x)= \sum _{l=0} ^{n-1}f'(l), $$

(1.7)

where $f_{n}(x)=f(x+n)$ ($n \in\mathbb{N}$); see [1, 9, 10, 15, 17].

Thus, by (1.6) we get

$$ \int_{\mathbb{Z}_{p}}e^{(x+y)t}\,du_{0}(y) = \frac{t}{e^{t}-1}e^{xt} = \sum_{n=0} ^{\infty}B_{n}(x)\frac{t^{n}}{n!}. $$

(1.8)

The modified degenerate Bernoulli polynomials are recently revisited by Dolgy et al., and they are formulated with the p-adic invariant integral on $\mathbb{Z}_{p}$ to be

$$\begin{aligned} \int_{\mathbb{Z}_{p}}(1+\lambda)^{(\frac{x+y}{\lambda})t}\,du_{0}(x) & = \frac{t}{(1+\lambda)^{\frac{t}{\lambda}} -1} \biggl( \frac{\log(1+ \lambda)}{\lambda} \biggr) (1+\lambda)^{\frac{xt}{\lambda}} \\ & = \sum_{n=0} ^{\infty} \beta_{n,\lambda}(x) \frac{t^{n}}{n!} \quad \bigl(\mbox{see [1]}\bigr), \end{aligned}$$

(1.9)

where $\lambda\in{\mathbb{C}}_{p}$ with $\vert \lambda \vert _{p}< p^{- \frac{1}{p-1}}$.

When $x=0$, we call $\beta_{n,\lambda}(0) = \beta_{n,\lambda}$ the modified degenerate Bernoulli numbers.

Recently, Kim introduced p-adic q-integral on $\mathbb{Z}_{p}$ is defined by

$$ \begin{aligned} I_{q}(f) & = \int_{\mathbb{Z}_{p}} f(x)\,d\mu_{q}(x)\\ & = \lim_{N \rightarrow \infty} \frac{1}{[p^{N}]_{q}} \sum_{x=0}^{p^{N} -1} f(x)q^{x} \quad (\text{see [17]}). \end{aligned} $$

(1.10)

The degenerate q-Bernoulli polynomials are also defined by Kim as follows.

$$ \sum_{n=0} ^{\infty} \beta_{n,q,\lambda}(x)\frac{t^{n}}{n!} = \int_{\mathbb{Z}_{p}}(1+\lambda t)^{\frac{[x+y]_{q}}{\lambda}}\,d\mu_{q}(y) \quad (\text{see [20]}). $$

(1.11)

The generating functions of Stirling numbers are given by

$$ \bigl(\log(1+t)\bigr)^{n} = n! \sum _{l=n} ^{\infty}S_{1}(l,n) \frac{t^{l}}{l!}\quad ( n \geq0 ) $$

(1.12)

and

$$ \bigl(e^{t} -1\bigr)^{n} = n! \sum _{l=n} ^{\infty}S_{2}(l,n) \frac{t^{l}}{l!} \quad ( n \geq0 ), $$

(1.13)

where $S_{1}(l,n)$ are the Stirling numbers of the first kind, and $S_{2}(l,n)$ are the Stirling numbers of the second kind.

The following diagram illustrates the variations of several types of q-Bernoulli polynomials and numbers. The definitions of the q-Bernoulli polynomials and the degenerate q-Bernoulli polynomials applied in the given diagram are provided by Carlitz [4, 5, 8] and Kim [20], respectively. In this paper, we investigate some of the explicit identities to characterize the modified degenerate q-Bernoulli polynomials used in the diagram

A few studies have identified some of the properties of the degenerate q-Bernoulli polynomials and numbers. This paper defines the modified q-Bernoulli polynomials and numbers arising from the p-adic invariant integral on $\mathbb{Z}_{p}$ and introduces additional characteristic properties of these polynomials and numbers, which are defined from the generating functions and p-adic invariant integral on $\mathbb{Z}_{p}$.

2 The modified degenerate q-Bernoulli polynomials and numbers

In the following discussions, we assume that $\lambda,t\in{\mathbb{C}} _{p}$ with $0 < \vert \lambda \vert \leq1$ and $\vert t\vert _{p} < p^{-\frac{1}{p-1}}$. Then, as $\vert \lambda t\vert _{p} < p^{-\frac{1}{p-1}}$, $\vert \log(1+ \lambda t)\vert _{p} = \vert \lambda t\vert _{p}$, and hence $\vert \frac{1}{\lambda}\log(1+\lambda t)\vert _{p} = \vert t\vert _{p} < p^{-\frac{1}{p-1}}$, it makes sense to take the limit as $\lambda\rightarrow0$.

Following (1.3), we define the modified degenerate q-Bernoulli polynomials given by the generating function

$$ \int_{\mathbb{Z}_{p}} q^{-y} (1+\lambda)^{\frac{[x+y]_{q}}{\lambda}t}\,du_{q}(y) = \sum_{n=0} ^{\infty} \widetilde{B}_{n,q,\lambda }(x)\frac{t ^{n}}{n!}. $$

(2.1)

When $x=0$, $\widetilde{B}_{n,q,\lambda}(0) = \widetilde{B}_{n,q, \lambda}$ are called the modified degenerate q-Bernoulli numbers.

Note that

$$ \begin{aligned}[b] & \lim_{\lambda\rightarrow0} \int_{\mathbb{Z}_{p}} q^{-y} (1+\lambda)^{\frac{[x+y]_{q}}{\lambda}t}\,du_{q}(y) \\ &\quad = \int_{\mathbb{Z}_{p}} q^{-y} e^{[x+y]_{q} t}\,du_{q}(y) \\ &\quad = \sum_{n=0} ^{\infty} B_{n,q}(x) \frac{t^{n}}{n!}, \end{aligned} $$

(2.2)

where $B_{n,q}(x)$ are the modified Carlitz q-Bernoulli polynomials.

Now, we consider

$$ \begin{aligned}[b] & \int_{\mathbb{Z}_{p}} q^{-y} (1+\lambda)^{\frac {[x+y]_{q}}{\lambda }t}\,du_{q}(y) \\ &\quad = \int_{\mathbb{Z}_{p}} q^{-y} e^{\frac{[x+y]_{q}}{\lambda} t \log(1+\lambda) }\,du_{q}(y) \\ &\quad = \sum_{n=0} ^{\infty} \biggl( \frac{\log(1+\lambda)}{\lambda } \biggr) ^{n} \int_{\mathbb{Z}_{p}}q^{-y} [x+y]_{q}^{n}\,du_{q}(y) \frac{t^{n}}{n!} \\ &\quad = \sum_{n=0} ^{\infty} \biggl( \frac{\log(1+\lambda)}{\lambda } \biggr) ^{n} B_{n,q}(x) \frac{t^{n}}{n!}. \end{aligned} $$

(2.3)

By the definitions provided in (2.1), (2.2), and (2.3) we are able to derive the following theorem.

Theorem 2.1

For $n\geq0$, $\widetilde{B}_{n,q,\lambda}(x)$ can be written as

$$ \widetilde{B}_{n,q,\lambda}(x) = \biggl( \frac{\log(1+\lambda)}{ \lambda} \biggr) ^{n} B_{n,q}(x). $$

(2.4)

Note that $(x)_{n} = \sum_{l=0} ^{n} S_{1} (n, l) x^{l}$ ($n \geq0$), where $S_{1}$ are the Stirling numbers of the first kind.

Then, by using (2.1) we are able to state

$$\begin{aligned}& \int_{\mathbb{Z}_{p}} q^{-y} (1+\lambda)^{\frac {[x+y]_{q}}{\lambda }t}\,du_{q}(y) \\& \quad = \sum_{n=0}^{\infty} \int_{\mathbb{Z}_{p}} q^{-y} \binom{\frac {[x+y]_{q}}{ \lambda}t}{n} \lambda^{n}\,du_{q}(y) \\& \quad = \sum_{n=0}^{\infty} \int_{\mathbb{Z}_{p}} q^{-y}\lambda^{n} \sum _{l=0}^{n} S_{1}(n,l) \biggl( \frac{[x+y]_{q}}{\lambda} \biggr) ^{l} \frac{t ^{l}}{n!}\,du_{q}(y) \\ & \quad = \sum_{l=0}^{\infty} \sum _{n=l}^{\infty} S_{1}(n,l) \lambda^{n-l} \frac{t^{l}}{n!} \int_{\mathbb{Z}_{p}} q^{-y} [x+y]_{q}^{l}\,du_{q}(y) \\ & \quad = \sum_{l=0}^{\infty} \Biggl( \sum _{n=l}^{\infty} S_{1}(n,l) \lambda^{n-l} \frac{l!}{n!} B_{l,q}(x) \Biggr) \frac{t^{l}}{l!}. \end{aligned}$$

(2.5)

Given the descriptions in (2.1) and (2.5), we have another theorem.

Theorem 2.2

For $n\geq0$, $\widetilde{B}_{n,q,\lambda}(x)$ can be written as

$$ \widetilde{B}_{n,q,\lambda}(x) = \sum_{n=l}^{\infty} S_{1}(n,l) \lambda^{n-l} \frac{l!}{n!} B_{l,q}(x). $$

(2.6)

We observe that

$$ \begin{aligned}[b] & \int_{\mathbb{Z}_{p}} q^{-y} (1+\lambda)^{\frac {[x+y]_{q}}{\lambda }t}\,du_{q}(y) \\ &\quad = \int_{\mathbb{Z}_{p}} q^{-y} (1+\lambda)^{\frac{[x]_{q}}{\lambda } t} (1+ \lambda)^{\frac{[y]_{q}}{\lambda} q^{x} t}\,du_{q}(y) \\ &\quad = (1+\lambda)^{\frac{[x]_{q}}{\lambda}t} \int_{\mathbb{Z}_{p}} q ^{-y} (1+\lambda)^{\frac{[y]_{q}}{\lambda} q^{x} t}\,du_{q}(y) \\ &\quad = \Biggl( \sum_{l=0}^{\infty} \biggl( \frac{\log(1+\lambda)}{ \lambda} \biggr) ^{l} [x]_{q}^{l} \frac{t^{l}}{l!} \Biggr) \Biggl( \sum_{m=0}^{\infty} \widetilde{B}_{m,q,\lambda} \frac{q^{mx}t^{m}}{m!} \Biggr) \\ &\quad = \sum_{n=0}^{\infty} \Biggl( \sum _{m=0}^{n} \binom{n}{m} \widetilde{B}_{m,q,\lambda}[x]_{q}^{n-m} q^{mx} \biggl( \frac{\log(1+ \lambda)}{\lambda} \biggr) ^{n-m} \Biggr) \frac{t^{n}}{n!}. \end{aligned} $$

(2.7)

The third theorem is obtained by (2.1) and (2.7) as follows.

Theorem 2.3

For $n\geq0$, $\widetilde{B}_{n,q,\lambda}(x)$ can be written as

$$ \widetilde{B}_{n,q,\lambda}(x) = \sum_{m=0}^{n} \binom{n}{m} \widetilde{B}_{m,q,\lambda}[x]_{q}^{n-m} q^{mx} \biggl( \frac{\log(1+ \lambda)}{\lambda} \biggr) ^{n-m}. $$

(2.8)

Remark 2.4

$$ \begin{aligned}[b] \lim_{\lambda\rightarrow0}\widetilde{B}_{m,q,\lambda}(x) &= \lim_{\lambda\rightarrow0} \sum_{m=0}^{n} \binom{n}{m} \widetilde{B}_{m,q,\lambda}[x]_{q}^{n-m} q^{mx} \biggl( \frac{\log(1+ \lambda)}{\lambda} \biggr) ^{n-m} \\ & = \sum_{m=0}^{n} \binom{n}{m} \widetilde{B}_{m,q} q^{mx} \\ & = B_{m,q}(x). \end{aligned} $$

(2.9)

Note that

$$\begin{aligned}& \int_{\mathbb{Z}_{p}} q^{-y} (1+\lambda)^{\frac {[x+y]_{q}}{\lambda }t}\,du_{q}(y) \\ & \quad = \lim_{N \rightarrow\infty} \frac{1}{[dp^{N}]_{q}} \sum _{y=0} ^{dp^{N} -1} (1+\lambda)^{\frac{[x+y]_{q}}{\lambda}t} \\& \quad = \lim_{N \rightarrow\infty} \frac{1}{[dp^{N}]_{q}} \sum _{a=0} ^{d-1} \sum_{y=0}^{p^{N} -1} (1+\lambda)^{\frac {[x+a+dy]_{q}}{\lambda }t} \\& \quad = \lim_{N \rightarrow\infty} \frac{1}{[d]_{q} [p^{N}]_{q^{d}}} \sum _{a=0}^{d-1} \sum_{y=0}^{p^{N} -1} (1+\lambda)^{\frac {1}{\lambda }[d]_{q} [\frac{x+a}{d} + y]_{q^{d}} t} \\& \quad = \frac{1}{[d]_{q}} \sum_{a=0}^{d-1} \lim _{N \rightarrow\infty} \frac{1}{[p ^{N}]_{q^{d}}} \sum_{y=0}^{p^{N} -1} (1+\lambda)^{\frac{1}{\lambda}[d]_{q} [\frac{x+a}{d} + y]_{q^{d}} t} q^{-dy} q^{dy} \\& \quad = \frac{1}{[d]_{q}} \sum_{a=0}^{d-1} \biggl( \int_{\mathbb{Z}_{p}} q ^{-dy} (1+\lambda)^{\frac{1}{\lambda}[d]_{q} [\frac{x+a}{d} + y]_{q ^{d}}t}\,du_{q^{d}}(y) \biggr) \\& \quad = \frac{1}{[d]_{q}} \sum_{a=0}^{d-1} \sum _{n=0}^{\infty} \widetilde{B}_{n,q^{d},\lambda} \biggl(\frac{x+a}{d}\biggr)\frac{[d]_{q}^{n} t ^{n}}{n!} \\& \quad = \sum_{n=0}^{\infty} \Biggl( [d]_{q}^{n-1} \sum_{a=0}^{d-1} \widetilde{B}_{n,q^{d},\lambda}\biggl(\frac{x+a}{d}\biggr) \Biggr) \frac{t^{n}}{n!}, \end{aligned}$$

(2.10)

where $d \in\mathbb{N}$.

The following theorem is obtained from (2.10).

Theorem 2.5

For $n\geq0$ and $d \in\mathbb{N}$, $\widetilde{B}_{n,q,\lambda}(x)$ can be written as

$$ \widetilde{B}_{n,q,\lambda}(x) = [d]_{q}^{n-1} \sum _{a=0}^{d-1} \widetilde{B}_{n,q^{d},\lambda} \biggl(\frac{x+a}{d}\biggr). $$

(2.11)

Now, we observe that

$$ \int_{\mathbb{Z}_{p}} q^{-y} e^{[x+y]_{q} t}\,du_{q}(y) = \sum_{n=0} ^{\infty} B_{n,q}(x) \frac{t^{n}}{n!}. $$

(2.12)

We obtain Theorem 2.1 as follows by substituting t by $\log(1+ \lambda)^{\frac{t}{\lambda}}$ in (2.12):

$$ \begin{aligned}[b] \int_{\mathbb{Z}_{p}} q^{-y} e^{[x+y]_{q} \log(1+ \lambda)^{\frac{t}{ \lambda}}}\,du_{q}(y) &= \int_{\mathbb{Z}_{p}} q^{-y} (1+\lambda)^{\frac{[x+y]_{q}}{ \lambda}t}\,du_{q}(y) \\ &= \sum_{n=0}^{\infty} B_{n,q}(x) \frac{1}{n!} \bigl( \log(1+ \lambda)^{\frac{t}{\lambda}} \bigr) ^{n} \\ &= \sum_{n=0}^{\infty} B_{n,q}(x) \biggl( {\frac{\log(1+ \lambda)}{ \lambda}} \biggr) ^{n} \frac{t^{n}}{n!}. \end{aligned} $$

(2.13)

For $r \in{\mathbb{N}}$, we define the modified degenerate q-Bernoulli polynomials of order r as follows:

$$\begin{aligned}& \int_{\mathbb{Z}_{p}}\cdots \int_{\mathbb{Z}_{p}} q^{-(x_{1}+x_{2}+ \cdots+ x_{r})} (1+\lambda)^{\frac{[x_{1}+x_{2}+ \cdots+ x_{r} +x]_{q}}{ \lambda} t}\,du_{q}(x_{1})\,du_{q}(x_{2})\cdots \,du_{q}(x_{r}) \\& \quad = \sum_{n=0} ^{\infty} \widetilde{B}_{n,q,\lambda}^{(r)}(x) \frac{t ^{n}}{n!}. \end{aligned}$$

(2.14)

When $x=0$, $\widetilde{B}_{n,q,\lambda}^{(r)}(0) = \widetilde{B} _{n,q,\lambda}^{(r)}$ are called the modified degenerate q-Bernoulli numbers of order r.

We observe that

$$ \begin{aligned}[b] & \int_{\mathbb{Z}_{p}}\cdots \int_{\mathbb{Z}_{p}} q^{-(x_{1}+x_{2}+ \cdots+ x_{r})} (1+\lambda)^{\frac{[x_{1}+x_{2}+ \cdots+ x_{r} +x]_{q}}{ \lambda}t}\,du_{q}(x_{1})\,du_{q}(x_{2})\cdots \,du_{q}(x_{r}) \\ &\quad = \int_{\mathbb{Z}_{p}}\cdots \int_{\mathbb{Z}_{p}} q^{-(x_{1}+x _{2}+ \cdots+ x_{r})} \sum_{n=0}^{\infty} \biggl( {\frac{\log(1+ \lambda)}{\lambda}} \biggr) ^{n} \\ &\qquad {} \times[x_{1}+x_{2}+ \cdots+ x_{r} +x]_{q}^{n} \frac{t^{n}}{n!}\,du _{q}(x_{1})\,du_{q}(x_{2}) \cdots \,du_{q}(x_{r}) \\ &\quad = \sum_{n=0}^{\infty} \biggl( { \frac{\log(1+ \lambda)}{\lambda }} \biggr) ^{n} \int_{\mathbb{Z}_{p}}\cdots \int_{\mathbb{Z}_{p}} q^{-(x_{1}+x _{2}+ \cdots+ x_{r})}[x_{1}+x_{2}+ \cdots+ x_{r} +x]_{q}^{n} \\ &\qquad {} \times du_{q}(x_{1})\,du_{q}(x_{2}) \cdots \,du_{q}(x_{r}) \frac{t^{n}}{n!} \\ &\quad = \sum_{n=0}^{\infty} \biggl( \biggl( { \frac{\log(1+ \lambda)}{ \lambda}} \biggr) ^{n} B_{n,q}^{(r)}(x) \biggr) \frac{t^{n}}{n!}. \end{aligned} $$

(2.15)

Therefore, we are able to derive the following theorem.

Theorem 2.6

For $n\geq0$, $\widetilde{B}_{n,q,\lambda}^{(r)}(x)$ can be written as

$$ \widetilde{B}_{n,q,\lambda}^{(r)}(x) = \biggl( {\frac{\log(1+ \lambda)}{\lambda}} \biggr) ^{n} B_{n,q}^{(r)}(x). $$

(2.16)

Now, we consider

$$\begin{aligned}& \int_{\mathbb{Z}_{p}}\cdots \int_{\mathbb{Z}_{p}} q^{-(x_{1}+x_{2}+ \cdots+ x_{r})} (1+\lambda)^{\frac{[x_{1}+x_{2}+ \cdots+ x_{r} +x]_{q}}{ \lambda}t}\,du_{q}(x_{1})\,du_{q}(x_{2})\cdots \,du_{q}(x_{r}) \\& \quad = \int_{\mathbb{Z}_{p}}\cdots \int_{\mathbb{Z}_{p}} q^{-(x_{1}+x _{2}+ \cdots+ x_{r})} \sum_{l=0}^{\infty} \binom{\frac{[x_{1}+x_{2}+ \cdots+ x_{r} +x]_{q}}{\lambda}t}{l} \lambda^{l}\,du_{q}(x_{1})\,du _{q}(x_{2})\cdots \,du_{q}(x_{r}) \\& \quad = \sum_{l=0}^{\infty}\sum _{n=0}^{l} \frac{S_{1}(l,n)}{l!} \lambda^{l-n} t^{n} \int_{\mathbb{Z}_{p}}\cdots \int_{\mathbb{Z}_{p}} q^{-(x_{1}+x_{2}+ \cdots+ x_{r})} \\& \qquad {} \times[x_{1}+x_{2}+ \cdots+ x_{r} +x]_{q}^{n}\,du_{q}(x_{1})\,du _{q}(x_{2})\cdots \,du_{q}(x_{r}) \\& \quad = \sum_{l=0}^{\infty}\sum _{n=0}^{l} \frac{S_{1}(l,n)}{l!} \lambda^{l-n} t^{n} B_{n,q}^{(r)}(x) \\& \quad = \sum_{n=0}^{\infty} \Biggl( \sum _{l=n}^{\infty} \frac{S_{1}(l,n)}{l!} \lambda^{l-n} n! B_{n,q}^{(r)}(x) \Biggr) \frac{t ^{n}}{n!}. \end{aligned}$$

(2.17)

Now, (2.17) yields the following theorem.

Theorem 2.7

For $n\geq0$, $\widetilde{B}_{n,q,\lambda}^{(r)}(x)$ can be written as

$$ \widetilde{B}_{n,q,\lambda}^{(r)}(x) = \sum _{l=n}^{\infty} \frac{S _{1}(l,n)}{l!} \lambda^{l-n} n! B_{n,q}^{(r)}(x). $$

(2.18)

Now, we observe that, for $d \in\mathbb{N}$,

$$\begin{aligned}& \int_{\mathbb{Z}_{p}}\cdots \int_{\mathbb{Z}_{p}} q^{-(x_{1}+x_{2}+ \cdots+ x_{r})} (1+\lambda)^{\frac{[x_{1}+x_{2}+ \cdots+ x_{r} +x]_{q}}{ \lambda}t}\,du_{q}(x_{1})\,du_{q}(x_{2})\cdots \,du_{q}(x_{r}) \\& \quad = \lim_{N \rightarrow\infty} \frac{1}{[dp^{N}]_{q}^{r}} \sum _{x_{1}=0}^{dp^{N} -1}\cdots\sum_{x_{r}=0}^{dp^{N} -1} (1+\lambda)^{\frac{[x_{1}+x_{2}+ \cdots+ x_{r} +x]_{q}}{\lambda}t} \\& \quad = \lim_{N \rightarrow\infty} \frac{1}{[dp^{N}]_{q}^{r}} \sum _{a_{1}=0}^{d-1}\cdots\sum_{a_{r}=0}^{d-1} \sum_{x_{1}=0}^{dp ^{N} -1}\cdots\sum _{x_{r}=0}^{dp^{N} -1} (1+\lambda)^{\frac{[a_{1} + \cdots+ a_{r} + x + dx_{1}+ d x_{2}+ \cdots+ d x_{r}]_{q}}{\lambda }t} \\& \quad = \lim_{N \rightarrow\infty} \frac{1}{[d]_{q}^{r} [p^{N}]_{q^{d}} ^{r}} \sum _{a_{1}=0}^{d-1}\cdots\sum_{a_{r}=0}^{d-1} \sum_{x_{1}=0} ^{p^{N} -1}\cdots\sum _{x_{r}=0}^{p^{N} -1} (1+\lambda)^{\frac{1}{ \lambda}[d]_{q} [\frac{a_{1} + \cdots+ a_{r} + x}{d} + x_{1}+x_{2}+ \cdots+ x_{r}]_{q^{d}} t} \\& \quad = \frac{1}{[d]_{q}^{r}} \sum_{a_{1}=0}^{d-1} \cdots\sum_{a_{r}=0} ^{d-1} \lim _{N \rightarrow\infty} \frac{1}{[p^{N}]_{q^{d}}^{r}} \sum_{x_{1}=0}^{p^{N} -1} \cdots\sum_{x_{r}=0}^{p^{N} -1} (1+\lambda)^{\frac{1}{\lambda}[\frac{a_{1} + \cdots+ a_{r} + x}{d} + x_{1}+x _{2}+ \cdots+ x_{r}]_{q^{d}} [d]_{q} t} \\& \quad = \frac{1}{[d]_{q}^{r}} \sum_{a_{1}=0}^{d-1} \cdots\sum_{a_{r}=0} ^{d-1} \int_{\mathbb{Z}_{p}} \cdots \int_{\mathbb{Z}_{p}} q^{-d(x_{1}+x _{2}+ \cdots+ x_{r})} \\& \qquad {} \times(1+\lambda)^{\frac{1}{\lambda}[\frac{a_{1} + \cdots+ a _{r} + x}{d} + x_{1}+x_{2}+ \cdots+ x_{r}]_{q^{d}} [d]_{q} t}\,du_{q ^{d}}(x_{1})\,du_{q^{d}}(x_{2}) \cdots \,du_{q^{d}}(x_{r}) \\& \quad = \sum_{n=0}^{\infty} \Biggl( [d]_{q}^{n-r} \sum_{a_{1}=0}^{d-1} \cdots\sum_{a_{r}=0}^{d-1} \widetilde{B}_{n,q^{d},\lambda}^{(r)} \biggl( \frac{a_{1} + \cdots+ a_{r} + x}{d} \biggr) \Biggr) \frac{t^{n}}{n!}. \end{aligned}$$

(2.19)

Finally, by comparing the coefficients on both sides of (2.19) we get the following theorem.

Theorem 2.8

For $n\geq0$ and $d \in\mathbb{N}$, $\widetilde{B}_{n,q}^{(r)}(x)$ can be written as

$$ \widetilde{B}_{n,q,\lambda}^{(r)}(x) = [d]_{q}^{n-r} \sum_{a_{1}=0} ^{d-1}\cdots\sum _{a_{r}=0}^{d-1} \widetilde{B}_{n,q^{d},\lambda} ^{(r)} \biggl( \frac{a_{1} + \cdots+ a_{r} + x}{d} \biggr). $$

(2.20)