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A note on \((h,q)\)-Boole polynomials

Advances in Difference Equations20152015:198

https://doi.org/10.1186/s13662-015-0536-1

  • Received: 19 April 2015
  • Accepted: 10 June 2015
  • Published:

Abstract

Kim et al. (Appl. Math. Inf. Sci. 9(6):1-6, 2015) consider the q-extensions of Boole polynomials. In this paper, we consider Witt-type formula for the q-Boole polynomials with weights and derive some new interesting identities and properties of those polynomials and numbers from the Witt-type formula which are related to special polynomials and numbers.

Keywords

  • \((h,q)\)-Euler polynomials
  • \((h,q)\)-Boole numbers and polynomials
  • p-adic invariant integral on \({{\mathbb{Z}}_{p}}\)

1 Introduction

Let p be a fixed odd prime number. Throughout this paper, \(\mathbb {Z}_{p}\), \(\mathbb{Q}_{p}\), and \(\mathbb{C}_{p}\) will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the completions of algebraic closure of \(\mathbb{Q}_{p}\). The p-adic norm is defined by \(|p|_{p} =\frac{1}{p}\).

When one talks of q-extension, q is variously considered as an indeterminate, a complex \(q \in\mathbb{C}\), or p-adic number \(q \in\mathbb{C}_{p}\). If \(q \in\mathbb{C}\), one normally assumes that \(|q|<1\). If \(q \in\mathbb{C}_{p}\), then we assume that \(|q-1|_{p} < p^{- \frac{1}{p-1}}\) so that \(q^{x} = \exp(x\log q)\) for each \(x \in\mathbb{Z}_{p}\). Throughout this paper, we use the notation
$$ [x]_{-q} =\frac{1- (-q)^{x}}{1-(-q)}. $$
Note that \(\lim_{q \rightarrow-1} [x]_{-q} = x \) for each \(x \in \mathbb{Z}_{p}\).
Let \(UD(\mathbb{Z}_{p})\) be the space of uniformly differentiable functions on \(\mathbb{Z}_{p}\). For \(f\in UD({\mathbb{Z}_{p}})\), the p-adic invariant integral on \({\mathbb{Z}_{p}}\) is defined by Kim as follows:
$$ I_{-q}(f)= \int_{\mathbb{Z}_{p}}f(x)\,d\mu_{-q}(x) =\lim_{N \rightarrow\infty} \frac{1}{[p^{N}]_{-q}}\sum _{x=0}^{p^{N}-1} f(x) (-q)^{x}\quad\mbox{(see [1--5])}. $$
(1.1)
Let \(f_{1}\) be the translation of f with \(f_{1} (x )=f (x+1 )\). Then, by (1.1), we get
$$ I_{-q}(f_{1})+I_{-q} (f)=[2]_{q} f(0). $$
(1.2)
As is well known, the Stirling number of the first kind is defined by
$$(x )_{n}=x (x-1 )\cdots (x-n+1 )=\sum _{l=0}^{n}S_{1} (n,l )x^{l}, $$
(1.3)
and the Stirling number of the second kind is given by the generating function:
$$\bigl(e^{t}-1 \bigr)^{m}=m!\sum _{l=m}^{\infty}S_{2} (l,m )\frac {t^{l}}{l!}\quad \mbox{(see [6, 7])}. $$
(1.4)
It is well known that the \((h,q)\) -Euler polynomials are defined by the generating function:
$$ \biggl(\frac{q+1}{q^{h}e^{t}+1} \biggr)e^{xt}=\sum _{n=0}^{\infty}E_{n,q} (x|h)\frac{t^{n}}{n!}\quad\mbox{(see [8])}, $$
(1.5)
where h is an integer. When \(x=0\) and \(h=0\), \(E_{n,q} (0|h) =E_{n,q}(h)\) are called the ordinary q-Euler numbers.
Recently, DS Kim and T Kim introduced the Changhee polynomials of the first kind are defined by the generating function:
$$ \frac{2}{2+t}(1+t)^{x}=\sum _{n=0}^{\infty}Ch_{n}(x)\frac{t^{n}}{n!}\quad\mbox{(see [1, 9--11])}, $$
(1.6)
and T Kim et al. defined the q-Changhee polynomials as follows:
$$ \frac{[2]_{q}}{q(1+t)+1}(1+t)^{x}=\sum _{n=0} ^{\infty} Ch_{n,q}(x)\frac {t^{n}}{n!}\quad\mbox{(see [9, 11, 12])}. $$
(1.7)
As is well known, the Boole polynomials are defined by the generating function:
$$ \sum_{n=0} ^{\infty}Bl_{n}(x|\lambda) \frac{t^{n}}{n!}=\frac {1}{1+(1+t)^{\lambda}}\quad\mbox{(see [7, 13])}. $$
When \(\lambda=1\), \(2Bl_{n}(x|1)=Ch_{n}(x)\) are Changhee polynomials. In [11], Kim et al. consider the q-analog of Boole polynomials, and found some new and interesting identities related to special polynomials, and Y Do and D Lim investigated the properties of \((h,q)\)-Daehee numbers and polynomials, which are defined by
$$ \int _{\mathbb {Z}_{p}}q^{-hy}(x+y)_{n}\,d\mu_{q}(y)\quad\mbox{(see [14])}. $$

In this paper, we consider Witt-type formula for the q-Boole polynomials with weights and derive some new interesting identities and properties of those polynomials and numbers from the Witt-type formula which are related to special polynomials and numbers.

2 q-Analog of Boole polynomials with weight

In this section, we assume that \(t\in{\mathbb{C}}_{p}\) with \(|t|_{p}< p^{-\frac{1}{p-1}}\), \(\lambda\in{\mathbb{Z}}_{p}\) with \(\lambda \neq0\) and \(h\in{\mathbb{Z}}\). From (1.2), we have
$$ \int _{\mathbb {Z}_{p}}q^{(h-1)y}(1+t)^{x+\lambda y}\,d\mu_{-q}(y)=\frac {1+q}{q^{h}(1+t)^{\lambda}+1}(1+t)^{x} =\sum_{n=0} ^{\infty}[2]_{q}Bl_{n,q}(x|h, \lambda)\frac{t^{n}}{n!}, $$
(2.1)
where \(Bl_{n,q} (x|h,\lambda)\) are the \((h,q)\)-Boole polynomials which are defined by
$$ \frac{1}{q^{h}(1+t)^{\lambda}+1}(1+t)^{x}=\sum _{n=0} ^{\infty }Bl_{n,q}(x|h,\lambda) \frac{t^{n}}{n!}. $$
(2.2)
By (2.1), we can derive the following equation:
$$ \int _{\mathbb {Z}_{p}}q^{(h-1)y}\binom{x+\lambda y}{n}\,d\mu_{-q}= \frac{1+q}{n!}Bl_{n,q} (x|h,\lambda). $$
(2.3)
In the special case \(x=0\), \(Bl_{n,q} (0|h,\lambda)=Bl_{n,q}(h,\lambda)\) are called the \((h,q)\) -Boole numbers.
Note that
$$ \begin{aligned}[b] (1+t)^{x+\lambda y}&=e^{(x+\lambda y)\log(1+t)}\\ &=\sum_{n=0} ^{\infty}\frac{(x+\lambda y)^{n}}{n!}\bigl( \log(1+t)\bigr)^{n}\\ &=\sum_{n=0} ^{\infty}\frac{(x+\lambda y)^{n}}{n!}m!\sum _{m=n} ^{\infty }S_{1}(m,n) \frac{t^{m}}{m!}\\ &=\sum_{n=0} ^{\infty} \Biggl\{ \sum _{m=0} ^{n} (x+\lambda y)^{m}S_{1}(n,m) \Biggr\} \frac{t^{n}}{n!}. \end{aligned} $$
(2.4)
The \((h,q)\) -Euler polynomials are defined by the generating function:
$$ \frac{1+q}{q^{h}e^{t}+1}e^{xt}=\sum _{n=0} ^{\infty}E_{n,q}(x|h)\frac{t^{n}}{n!}. $$
(2.5)
Note that \(\lim_{q\rightarrow1}E_{n,q}(x|1)=E_{n}(x)\). When \(x=0\), \(E_{n}(0|h)=E_{n,q}(h)\) are called the \((h,q)\) -Euler numbers.
By (1.2), we can derive easily the following equation:
$$ \int _{\mathbb {Z}_{p}}q^{(h-1)y}e^{(x+y)t}\,d\mu_{-q}(y)=\frac{1+q}{q^{h}e^{t}+1}e^{xt} =\sum_{n=0} ^{\infty}E_{n,q} (x|h) \frac{t^{n}}{n!}. $$
(2.6)
Since
$$ \int _{\mathbb {Z}_{p}}q^{(h-1)y}e^{(x+y)t}\,d\mu_{-q}(y)=\sum _{n=0} ^{\infty} \int _{\mathbb {Z}_{p}}q^{(h-1)y}(x+y)^{n} \,d \mu_{-q}(y)\frac{t^{n}}{n!}, $$
by (2.5), we have
$$ \int _{\mathbb {Z}_{p}}q^{(h-1)y}(x+y)^{n}\,d\mu_{-q}(y)=E_{n,q} (x|h) \quad(n\geq0). $$
(2.7)
From (2.4) and (2.7), we get
$$ \begin{aligned}[b] &\int _{\mathbb {Z}_{p}}q^{(h-1)y}(1+t)^{x+\lambda y}\,d\mu_{-q}(y)\\ &\quad=\sum_{n=0} ^{\infty} \Biggl\{ \sum _{m=0} ^{n} \int _{\mathbb {Z}_{p}}q^{(h-1)y}(x+\lambda y)^{m} \,d\mu_{-q}(y) S_{1}(n,m) \Biggr\} \frac{t^{n}}{n!}\\ &\quad=\sum_{n=0} ^{\infty} \Biggl\{ \sum _{m=0} ^{n}\lambda^{m} E_{m,q} \biggl(\frac{x}{\lambda} \vert h \biggr)S_{1}(n,m) \Biggr\} \frac{t^{n}}{n!}. \end{aligned} $$
(2.8)
Thus, by (2.2), (2.3), and (2.8), we obtain the following theorem.

Theorem 2.1

For \(n \geq0\), we have
$$ Bl_{n,q}(x|h,\lambda)=\frac{1}{[2]_{q}}\sum _{m=0} ^{n} \lambda^{m}E_{m,q} \biggl(\frac{x}{\lambda}\Big| h \biggr)S_{1}(n,m) $$
and
$$ \int _{\mathbb {Z}_{p}}q^{(h-1)y}\binom{x+\lambda y}{n}\,d\mu_{-q}=\frac{[2]_{q}}{n!}Bl_{n,q} (x|h,\lambda). $$
By Theorem 2.1, we note that
$$ Bl_{n,q}(x|h,\lambda)=\frac{1}{[2]_{q}}\int _{\mathbb {Z}_{p}}q^{(h-1)y}(x+\lambda y)_{n}\,d\mu_{-q}(y), $$
where \((x)_{n}=x(x-1)\cdots(x-n+1)\). When \(\lambda=1\) and \(h=0\), we have
$$ Bl_{n,q}(x|0,1)=\frac{1}{[2]_{q}}\int _{\mathbb {Z}_{p}}q^{-1}(x+y)^{n}\,d\mu_{-q}(y). $$
(2.9)
In [13], Arici et al. defined the q-analog of Changhee polynomials by the generating function:
$$ \sum_{n=0} ^{\infty}Ch_{n}(x|q) \frac{t^{n}}{n!}=\frac{[2]_{q}}{[2]_{t}+1}(1+t)^{x}. $$
(2.10)
By (2.10), we have
$$ \int _{\mathbb {Z}_{p}}q^{-y}(1+t)^{x+y}\,d\mu_{-q}(y)= \frac{[2]_{q}}{[2]_{t}+1}(1+t)^{x}=\sum_{n=0} ^{\infty}Ch_{n}(x|q)\frac{t^{n}}{n!}. $$
(2.11)
By (1.6) and (2.10), we note that
$$ \frac{[2]_{q}}{2}Ch_{n}(x)=Ch_{n}(x|q). $$
(2.12)
From (2.11), we get
$$ \int _{\mathbb {Z}_{p}}q^{-1}(x+y)_{n}\,d\mu_{-q}(y)=Ch_{n}(x|q). $$
(2.13)
By (2.9), (2.12), and (2.13), we have
$$ Bl_{n,q} (x|0,1)=\frac{1}{[2]_{q}}Ch_{n}(x|q)= \frac{1}{2}Ch_{n}(x). $$
By replacing t as \(e^{t}-1\) in (2.1), we derive the following equations:
$$ \begin{aligned}[b] \frac{1+q}{q^{h}e^{\lambda t}+1}e^{xt}&=\sum _{n=0} ^{\infty }[2]_{q}Bl_{n,q}(x|h, \lambda)\frac{1}{n!}\bigl(e^{t}-1\bigr)^{n}\\ &=\sum_{n=0} ^{\infty}[2]_{q}Bl_{n,q}(x|h, \lambda)\frac{1}{n!}n!\sum_{m=n} ^{\infty}S_{2}(m,n)\frac{t^{m}}{m!}\\ &=\sum_{n=0} ^{\infty}\sum _{m=0} ^{n} [2]_{q}Bl_{m,q}(x|h, \lambda )S_{2}(n,m)\frac{t^{n}}{n!} \end{aligned} $$
(2.14)
and
$$ \frac{1+q}{q^{h}e^{\lambda t}+1}e^{xt}= \frac{1+q}{q^{h}e^{\lambda t}+1}e^{ (\frac{x}{\lambda} )\lambda t} =\sum_{n=0} ^{\infty}E_{n,q} \biggl( \frac{x}{\lambda}\Big| h \biggr)\lambda^{m}\frac{t^{m}}{m!}. $$
(2.15)
Hence, by (2.14) and (2.15), we obtain the following theorem.

Theorem 2.2

For \(n \geq0\), we have
$$ \sum_{m=0} ^{n}Bl_{m,q}(x|h, \lambda)S_{2}(n,m)=\frac{\lambda ^{m}}{q+1}E_{n,q} \biggl( \frac{x}{\lambda}\Big| h \biggr). $$
From now on, we define the \((h_{1},\ldots,h_{r},q)\) -Boole numbers of the first kind as follows:
$$ \begin{aligned}[b] &[2]_{q} ^{r} Bl_{n,q} ^{(h_{1},\ldots,h_{r})} (\lambda)\\ &\quad=\int _{\mathbb {Z}_{p}}\cdots \int _{\mathbb {Z}_{p}}q^{h_{1}+\cdots+h_{r}-r}\bigl(\lambda(x_{1}+ \cdots+x_{r})\bigr)_{n}\,d\mu _{-q}(x_{1}) \cdots \,d\mu_{-q}(x_{r}) \quad(n \geq0). \end{aligned} $$
(2.16)
By (2.16), we have
$$\begin{aligned} &[2]_{q} ^{r} \sum _{n=0} ^{\infty}Bl_{n,q} ^{(h_{1},\ldots,h_{r})} (\lambda) \frac{t^{n}}{n!} \\ &\quad=\int _{\mathbb {Z}_{p}}\cdots \int _{\mathbb {Z}_{p}}\sum_{n=0} ^{\infty} q^{h_{1}+\cdots+h_{r}-r}\binom{\lambda (x_{1}+\cdots+x_{r})}{n}t^{n}\,d\mu_{-q}(x_{1}) \cdots \,d\mu_{-q}(x_{r}) \\ &\quad=\sum_{n=0} ^{\infty} \int _{\mathbb {Z}_{p}}\cdots \int _{\mathbb {Z}_{p}}q^{h_{1}+\cdots+h_{r}-r}(1+t)^{\lambda (x_{1}+\cdots+x_{k})}\,d\mu_{-q}(x_{1})\cdots \,d \mu_{-q}(x_{r}) \\ &\quad=\prod_{i=1} ^{r} \biggl( \frac{1+q}{q^{h_{i}}(1+t)^{\lambda}+1} \biggr) \\ &\quad=(1+q)^{r}\sum_{n=0} ^{\infty} \biggl(\sum_{l_{1}+\cdots+l_{r}=n}\binom {n}{l_{1},\ldots,l_{r}}B_{i_{1},q} (h, \lambda)\cdots B_{i_{r},q} (h,\lambda ) \biggr)\frac{t^{n}}{n!}. \end{aligned}$$
(2.17)
Thus, by (2.17), we obtain the following corollary.

Corollary 2.3

For \(n \geq0\), we have
$$ Bl_{n,q} ^{(h_{1},\ldots,h_{r})} (\lambda)=\sum_{l_{1}+\cdots+l_{r}=n} \binom {n}{l_{1},\ldots,l_{r}}B_{i_{1},q} (h,\lambda)\cdots B_{i_{r},q} (h,\lambda). $$
The \((h_{1},\ldots,h_{r},q)\) -Euler polynomials are defined by the generating function to be
$$\begin{aligned} &\int _{\mathbb {Z}_{p}}\cdots \int _{\mathbb {Z}_{p}}q^{h_{1}+\cdots+h_{r}-r}e^{(x_{1}+\cdots+x_{r}+x)t}\,d\mu _{-q}(x_{1})\cdots \,d\mu_{-q}(x_{r}) \\ &\quad=\prod_{i=1} ^{r} \biggl( \frac{1+q}{q^{h_{i}}e^{t}+1} \biggr)e^{xt} \\ &\quad=\sum_{n=0} ^{\infty}E_{n,q}(x|h_{1},\ldots,h_{r}) \frac{t^{n}}{n!}. \end{aligned}$$
(2.18)
By (2.18), we have
$$ \int _{\mathbb {Z}_{p}}\cdots \int _{\mathbb {Z}_{p}}q^{h_{1}+\cdots+h_{r}-r}(x_{1}+\cdots+x_{r}+x)^{n}\,d \mu _{-q}(x_{1})\cdots \,d\mu_{-q}(x_{r})=E_{n,q} (x|h_{1},\ldots,h_{r}). $$
In the special case \(x=0\), \(E_{n,q} (0|h_{1},\ldots,h_{r})=E_{n,q} (h_{1},\ldots,h_{r})\) are called the \((h_{1},\ldots,h_{r},q)\) -Euler numbers.
From (1.5) and (2.16), we note that
$$\begin{aligned} &(1+q)^{r}Bl_{n,q} ^{(h_{1},\ldots,h_{r})}(\lambda) \\ &\quad=\int _{\mathbb {Z}_{p}}\cdots \int _{\mathbb {Z}_{p}}q^{h_{1}+\cdots+h_{r}-r}\bigl(\lambda(x_{1}+ \cdots+x_{r})\bigr)_{n}\,d\mu _{-q}(x_{1}) \cdots \,d\mu_{-q}(x_{r}) \\ &\quad=\sum_{l=0} ^{n} S_{1}(n,l)\int _{\mathbb {Z}_{p}}\cdots \int _{\mathbb {Z}_{p}}q^{h_{1}+\cdots+h_{r}-r}\lambda ^{l}(x_{1}+ \cdots+x_{r})^{l}\,d\mu_{-q}(x_{1})\cdots \,d \mu_{-q}(x_{r}) \\ &\quad=\sum_{l=0} ^{n} S_{1}(n,l) \lambda^{l}E_{l,q}(h_{1},\ldots,h_{r}). \end{aligned}$$
(2.19)
Therefore, by (2.19), we obtain the following theorem.

Theorem 2.4

For \(n \geq0\), we get
$$ Bl_{n,q} ^{(h_{1},\ldots,h_{r})}(\lambda)=\frac{1}{(1+q)^{r}}\sum _{l=0} ^{n} S_{1}(n,l)\lambda^{l}E_{l,q}(h_{1}, \ldots,h_{r}). $$
By replacing t by \(e^{t}-1\) in (2.17), we have
$$\begin{aligned}{} [2]_{q} ^{r}\sum _{n=0} ^{\infty}Bl_{n,q} ^{(h_{1},\ldots,h_{r})}(\lambda)\frac {(e^{t}-1)^{n}}{n!}&=\prod_{i=1} ^{r} \biggl(\frac{1+q}{q^{h_{i}}e^{\lambda t}+1} \biggr) \\ &=\sum_{n=0} ^{\infty}E_{n,q} (h_{1},\ldots,h_{r})\lambda^{n}\frac{t^{n}}{n!} \end{aligned}$$
(2.20)
and
$$\begin{aligned}{} [2]_{q} ^{r}\sum _{n=0} ^{\infty}Bl_{n,q} ^{(h_{1},\ldots,h_{r})}(\lambda)\frac {1}{n!}\bigl(e^{t}-1 \bigr)^{n} &=[2]_{q} ^{r}\sum_{n=0} ^{\infty}Bl_{n,q} ^{(h_{1},\ldots,h_{r})}(\lambda)\sum _{m=n} ^{\infty}S_{2}(m,n)\frac{t^{m}}{m!} \\ &=[2]_{q} ^{r}\sum_{n=0} ^{\infty} \Biggl\{ \sum_{m=0} ^{n} Bl_{m,q} ^{(h_{1},\ldots,h_{r})}(\lambda)S_{2}(n,m) \Biggr\} \frac{t^{n}}{n!}. \end{aligned}$$
(2.21)
Hence, by (2.20) and (2.21), we obtain the following theorem.

Theorem 2.5

For \(n \geq0\), we have
$$ \frac{\lambda^{n}}{[2]_{q} ^{r}}E_{n,q} (h_{1},\ldots,h_{r})= \sum_{m=0} ^{n} Bl_{m,q} ^{(h_{1},\ldots,h_{r})}(\lambda)S_{2}(n,m). $$
Let us define the \((h_{1},\ldots,h_{r},q)\) -Boole polynomials of the first kind as follows:
$$\begin{aligned} &[2]_{q} ^{r}Bl_{n,q} ^{(h_{1},\ldots,h_{r})}(x|\lambda) \\ &\quad=\int _{\mathbb {Z}_{p}}\cdots \int _{\mathbb {Z}_{p}}q^{h_{1}+\cdots+h_{r}-r}\bigl(\lambda(x_{1}+ \cdots+x_{r})+x\bigr)_{n}\,d\mu _{-q}(x_{1}) \cdots \,d\mu_{-q}(x_{r}), \end{aligned}$$
(2.22)
where \(n \geq0\) and \(r\in{\mathbb{N}}\). By (2.22), we can derive the generating function of the \((h_{1},\ldots ,h_{r},q)\)-Boole polynomials of the first kind as follows:
$$\begin{aligned} &[2]_{q} ^{r}\sum _{n=0} ^{\infty}Bl_{n,q} ^{(h_{1},\ldots,h_{r})}(x|\lambda )\frac{t^{n}}{n!} \\ &\quad=\int _{\mathbb {Z}_{p}}\cdots \int _{\mathbb {Z}_{p}}q^{h_{1}+\cdots+h_{r}-r}(1+t)^{\lambda(x_{1}+\cdots+x_{r})+x}\,d\mu _{-q}(x_{1}) \cdots \,d\mu_{-q}(x_{r}) \\ &\quad=\prod_{i=1} ^{r} \biggl( \frac{1+q}{q^{h_{i}}(1+t)^{\lambda}+1} \biggr) (1+t)^{x}. \end{aligned}$$
(2.23)
By (2.23), we can see easily
$$\begin{aligned} &\prod_{i=1} ^{r} \biggl(\frac{1+q}{q^{h_{i}}(1+t)^{\lambda}+1} \biggr) (1+t)^{x} \\ &\quad=[2]_{q} ^{r} \Biggl(\sum_{n=0} ^{\infty}Bl_{n,q} ^{(h_{1},\ldots,h_{r})}(\lambda )\frac{t^{n}}{n!} \Biggr) \Biggl(\sum_{m=0} ^{\infty} \binom{x}{m}t^{m} \Biggr) \\ &\quad=[2]_{q} ^{r} \sum_{n=0} ^{\infty} \Biggl(\sum_{m=0} ^{n} m! \binom{x}{m}\frac {n!}{(n-m)!m!}Bl_{n-m,q} ^{(h_{1},\ldots,h_{r})}(\lambda) \Biggr)\frac {t^{n}}{n!} \\ &\quad=[2]_{q} ^{r} \sum_{n=0} ^{\infty} \Biggl(\sum_{m=0} ^{n} m! \binom{x}{m}\binom {n}{m}Bl_{n-m,q} ^{(h_{1},\ldots,h_{r})}(\lambda) \Biggr)\frac{t^{n}}{n!} \\ &\quad=[2]_{q} ^{r} \sum_{n=0} ^{\infty} \Biggl(\sum_{m=0} ^{n} \binom {n}{m}Bl_{n-m,q} ^{(h_{1},\ldots,h_{r})}(\lambda) (x)_{m} \Biggr)\frac{t^{n}}{n!}. \end{aligned}$$
(2.24)
By (2.23) and (2.24), we obtain the following theorem.

Theorem 2.6

For \(n \geq0\), we have
$$ Bl_{n,q} ^{(h_{1},\ldots,h_{r})}(x|\lambda)=\sum_{m=0} ^{n} \binom {n}{m}Bl_{n-m,q} ^{(h_{1},\ldots,h_{r})}(\lambda) (x)_{m}. $$
Replacing t as \(e^{t}-1\) in (2.23), we get
$$\begin{aligned}{} [2]_{q} ^{r}\sum _{n=0} ^{\infty}Bl_{n,q} ^{(h_{1},\ldots,h_{r})}(x|\lambda)\frac {1}{n!}\bigl(e^{t}-1 \bigr)^{n}&=\prod_{i=1} ^{n} \biggl(\frac{1+q}{q^{h_{i}}e^{\lambda t}+1} \biggr)e^{xt} \\ &=\sum_{n=0} ^{\infty}E_{n,q} ^{(h_{1},\ldots,h_{r})} \biggl(\frac{x}{\lambda } \biggr)\lambda^{n} \frac{t^{n}}{n!} \end{aligned}$$
(2.25)
and
$$\begin{aligned} &[2]_{q} ^{r}\sum _{n=0} ^{\infty}Bl_{n,q} ^{(h_{1},\ldots,h_{r})}(x|\lambda )\frac{(e^{t}-1)^{n}}{n!} \\ &\quad=[2]_{q} ^{r}\sum_{n=0} ^{\infty} \Biggl(\sum_{m=0} ^{n} Bl_{m,q} ^{(h_{1},\ldots ,h_{r})}(x|\lambda)S_{2}(n,m) \Biggr) \frac{t^{n}}{n!}. \end{aligned}$$
(2.26)
Hence, by (2.25) and (2.26), we obtain the following theorem.

Theorem 2.7

For \(n \geq0\), we have
$$ \sum_{m=0} ^{n} Bl_{m,q} ^{(h_{1},\ldots,h_{r})}(x|\lambda)S_{2}(n,m)=\frac {\lambda^{n}}{[2]_{q} ^{r}}E_{n,q} ^{(h_{1},\ldots,h_{r})} \biggl(\frac{x}{\lambda } \biggr). $$
From (2.23), we get
$$\begin{aligned} &[2]_{q} ^{r}Bl_{n,q} ^{(h_{1},\ldots,h_{r})}(x|\lambda) \\ &\quad=\int _{\mathbb {Z}_{p}}\cdots \int _{\mathbb {Z}_{p}}q^{h_{1}+\cdots+h_{r}-r}\bigl(\lambda(x_{1}+ \cdots+x_{r})+x\bigr)_{n}\,d\mu _{-q}(x_{1}) \cdots \,d\mu_{-q}(x_{r}) \\ &\quad=\sum_{l=0} ^{n} S_{1}(n,l) \int _{\mathbb {Z}_{p}}\cdots \int _{\mathbb {Z}_{p}}q^{h_{1}+\cdots+h_{r}-r}\bigl(\lambda(x_{1}+\cdots +x_{r})+x\bigr)^{l}\,d\mu_{-q}(x_{1}) \cdots \,d\mu_{-q}(x_{r}) \\ &\quad=\sum_{l=0} ^{n} S_{1}(n,l) \lambda^{l}E_{n,q} ^{(h_{1},\ldots,h_{r})} \biggl(\frac {x}{\lambda} \biggr). \end{aligned}$$
(2.27)
Thus, by (2.27), we obtain the following theorem.

Theorem 2.8

For \(n\geq0\), we have
$$ Bl_{n,q} ^{(h_{1},\ldots,h_{r})}(x|\lambda)=\frac{1}{[2]_{q} ^{r}}\sum _{l=0} ^{n} S_{1}(n,l)\lambda^{l}E_{n,q} ^{(h_{1},\ldots,h_{r})} \biggl(\frac{x}{\lambda } \biggr). $$
Now, we define the \((h,q)\) -Boole polynomials of the second kind as follows:
$$ {\widehat{Bl}}_{n,q}(x|h,\lambda)=\frac{1}{[2]_{q}}\int _{\mathbb {Z}_{p}}q^{(h-1)y}(-\lambda y+x)_{n}\,d\mu_{-q}(y) \quad(n \geq0). $$
(2.28)
By (2.28), we have
$$\begin{aligned} {\widehat{Bl}}_{n,q}(x|h, \lambda)&=\frac{1}{[2]_{q}}\sum_{l=0} ^{n}(- \lambda )^{l} S_{1}(n,l)\int _{\mathbb {Z}_{p}}\biggl(y-\frac{x}{\lambda} \biggr)^{l}\,d\mu_{-q}(y) \\ &=\frac{1}{[2]_{q}}\sum_{l=0} ^{n}(- \lambda)^{l} S_{1}(n,l)E_{l,q} \biggl(- \frac {x}{\lambda} \biggr). \end{aligned}$$
(2.29)
In the special case \(x=0\), \({\widehat{Bl}}_{n,q} (0|h,\lambda)={\widehat {Bl}}_{n,q}(h,\lambda)\) are called the \((h,q)\) -Boole numbers of the second kind. From (2.29), we can derive the generating function of \({\widehat{Bl}}_{n,q}(x|\lambda)\) as follows:
$$\begin{aligned} \sum_{n=0} ^{\infty}{\widehat{Bl}}_{n,q}(x|h,\lambda)\frac {t^{n}}{n!}&= \frac{1}{[2]_{q}}\int _{\mathbb {Z}_{p}}q^{(h-1)y}(1+t)^{-\lambda y+x}\,d\mu _{-q}(y) \\ &=\frac{(1+t)^{\lambda}}{q^{h}+(1+t)^{\lambda}}(1+t)^{x}. \end{aligned}$$
(2.30)
By replacing t by \(e^{t}-1\) in (2.30), we have
$$\begin{aligned} \sum_{n=0} ^{\infty}{\widehat{Bl}}_{n,q}(x|h,\lambda)\frac {(e^{t}-1)^{n}}{n!}&= \frac{e^{\lambda t}}{q^{h}+e^{\lambda t}}e^{xt} \\ &=\frac{1}{1+q}\sum_{n=0} ^{\infty}(- \lambda)^{n}E_{n,q} \biggl(-\frac {\lambda}{x}\Big| h \biggr)\frac{t^{n}}{n!} \end{aligned}$$
(2.31)
and
$$ \sum_{n=0} ^{\infty}{ \widehat{Bl}}_{n,q}(x|h,\lambda)\frac{(e^{t}-1)^{n}}{n!} =\sum _{n=0} ^{\infty} \Biggl(\sum_{m=0} ^{n} {\widehat {Bl}}_{m,q}(x|h,\lambda)S_{2}(n,m) \Biggr)\frac{t^{n}}{n!}. $$
(2.32)
By (2.31) and (2.32), we obtain the following theorem.

Theorem 2.9

For \(n \geq0\), we have
$$ {\widehat{Bl}}_{n,q}(x|h,\lambda)=\frac{1}{[2]_{q}}\sum _{l=0} ^{n}(-\lambda )^{l} S_{1}(n,l)E_{l,q} \biggl(-\frac{x}{\lambda} \biggr) $$
and
$$ \frac{1}{[2]_{q}}(-\lambda)^{n}E_{n,q} \biggl(- \frac{\lambda}{x}\Big| h \biggr)=\sum_{m=0} ^{n} {\widehat{Bl}}_{m,q}(x|h,\lambda)S_{2}(n,m). $$
For \(h_{1},\ldots,h_{r}\in{\mathbb{Z}}\), we define the \((h_{1},\ldots ,h_{r},q)\) -Boole polynomials of the second kind as follows:
$$\begin{aligned} &{\widehat{Bl}}_{n,q} ^{(h_{1},\ldots,h_{r})}(x|\lambda) \\ &\quad=\frac{1}{q+1}\int _{\mathbb {Z}_{p}}\cdots \int _{\mathbb {Z}_{p}}q^{(h_{1}+\cdots+h_{r}-r)y} \bigl(-\lambda (x_{1}+\cdots+x_{r})+x \bigr)_{n}\,d \mu_{-q}(x_{1})\cdots \,d\mu_{-q}(x_{r}). \end{aligned}$$
(2.33)
By (2.33), we can derive the generating function of the \((h_{1},\ldots ,h_{r},q)\)-Boole polynomials of the second kind as follows:
$$\begin{aligned} &\sum_{n=0} ^{\infty}{\widehat{Bl}}_{n,q} ^{(h_{1},\ldots,h_{r})}(x|\lambda ) \frac{t^{n}}{n!} \\ &\quad=\frac{1}{(1+q)^{r}}\int _{\mathbb {Z}_{p}}\cdots \int _{\mathbb {Z}_{p}}(1+t)^{-\lambda x_{1}-\cdots-\lambda x_{r}+x}\,d\mu_{-q}(x_{1}) \cdots \,d\mu_{-q}(x_{r}) \\ &\quad=\prod_{i=1} ^{r} \biggl( \frac{(1+t)^{\lambda}}{q^{h_{i}}+(1+t)^{\lambda }} \biggr) (1+t)^{x} \\ &\quad=\prod_{i=1} ^{r} \biggl( \frac{1}{q^{h_{i}}(1+t)^{-\lambda}+1} \biggr) (1+t)^{x} \\ &\quad=\sum_{n=0} ^{\infty}Bl_{n,q} ^{(h_{1},\ldots,h_{r})}(x|-\lambda)\frac{t^{n}}{n!}. \end{aligned}$$
(2.34)
Hence, by (2.34), we obtain the following proposition.

Proposition 2.10

For \(n \geq0\), we have
$$ {\widehat{Bl}}_{n,q} ^{(h_{1},\ldots,h_{r})}(x|\lambda)=Bl_{n,q} ^{(h_{1},\ldots,h_{r})}(x|-\lambda). $$
Note that
$$\begin{aligned} \frac{(-1)^{n}[2]_{q}}{n!}Bl_{n,q} (x|h, \lambda)&=(-1)^{n}\int _{\mathbb {Z}_{p}}q^{(h-1)y}\binom {x+\lambda y}{n}\,d\mu_{-q}(y) \\ &=\int _{\mathbb {Z}_{p}}q^{(h-1)y}\binom{-x-\lambda y+n-1}{n}\,d\mu_{-q}(y) \\ &=\int _{\mathbb {Z}_{p}}q^{(h-1)y}\sum_{m=0} ^{n} \binom{-x-\lambda y}{m}\binom{n-1}{n-m}\,d\mu _{-q}(y) \\ &=\sum_{m=0} ^{n} \binom{n-1}{n-m}\int _{\mathbb {Z}_{p}}q^{(h-1)y}\binom{-x-\lambda y}{m}\,d\mu_{-q}(y) \\ &=[2]_{q}\sum_{m=0} ^{n} \binom{n-1}{n-m}\frac{{\widehat{Bl}}_{m,q} (-x|h,\lambda)}{m!}, \end{aligned}$$
(2.35)
and, by a similar method, we get
$$\begin{aligned} \frac{(-1)^{n}[2]_{q}}{n!}{\widehat{Bl}}_{n,q} (x|h,\lambda)&=(-1)^{n}\int _{\mathbb {Z}_{p}}q^{(h-1)y}\binom{x-\lambda y}{n}\,d\mu_{-q}(y) \\ &=[2]_{q}\sum_{m=0} ^{n} \binom{n-1}{n-m}\frac{Bl_{m,q} (-x|h,\lambda)}{m!}. \end{aligned}$$
(2.36)
By (2.35) and (2.36), we obtain the following theorem.

Theorem 2.11

For \(n \geq0\), we have
$$ \frac{(-1)^{n}}{n!}Bl_{n,q} (x|h,\lambda)=\sum _{m=0} ^{n} \binom {n-1}{n-m}\frac{{\widehat{Bl}}_{m,q} (-x|h,\lambda)}{m!} $$
and
$$ \frac{(-1)^{n}}{n!}{\widehat{Bl}}_{n,q} (x|h,\lambda)=\sum _{m=0} ^{n} \binom {n-1}{n-m}\frac{Bl_{m,q} (-x|h,\lambda)}{m!}. $$

By Theorem 2.11, we obtain the following corollary.

Corollary 2.12

For \(n \geq0\), we have
$$ Bl_{n,q} (x|h,\lambda)=\sum_{m=0} ^{n} \sum_{k=0} ^{m} (-1)^{n+m}\binom {n}{n-m,m-k,k}(n-1)_{l-1}Bl_{k,q}(x|h, \lambda) $$
where \(\binom{n}{p,q,r}=\frac{n!}{p!q!r!}\), \(p+q+r=n\).

Declarations

Acknowledgements

The authors are grateful for the valuable comments and suggestions of the referees.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Kyungpook National University, Taegu, 702-701, Republic of Korea
(2)
Department of Mathematics Education, Kyungpook National University, Taegu, 702-701, Republic of Korea

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