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A note on \((h,q)\)-Boole polynomials
Advances in Difference Equations volume 2015, Article number: 198 (2015)
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.
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
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:
Let \(f_{1}\) be the translation of f with \(f_{1} (x )=f (x+1 )\). Then, by (1.1), we get
As is well known, the Stirling number of the first kind is defined by
and the Stirling number of the second kind is given by the generating function:
It is well known that the \((h,q)\) -Euler polynomials are defined by the generating function:
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:
and T Kim et al. defined the q-Changhee polynomials as follows:
As is well known, the Boole polynomials are defined by the generating function:
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
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
where \(Bl_{n,q} (x|h,\lambda)\) are the \((h,q)\)-Boole polynomials which are defined by
By (2.1), we can derive the following equation:
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
The \((h,q)\) -Euler polynomials are defined by the generating function:
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:
Since
by (2.5), we have
Thus, by (2.2), (2.3), and (2.8), we obtain the following theorem.
Theorem 2.1
For \(n \geq0\), we have
and
By Theorem 2.1, we note that
where \((x)_{n}=x(x-1)\cdots(x-n+1)\). When \(\lambda=1\) and \(h=0\), we have
In [13], Arici et al. defined the q-analog of Changhee polynomials by the generating function:
By (2.10), we have
By (1.6) and (2.10), we note that
From (2.11), we get
By (2.9), (2.12), and (2.13), we have
By replacing t as \(e^{t}-1\) in (2.1), we derive the following equations:
and
Hence, by (2.14) and (2.15), we obtain the following theorem.
Theorem 2.2
For \(n \geq0\), we have
From now on, we define the \((h_{1},\ldots,h_{r},q)\) -Boole numbers of the first kind as follows:
By (2.16), we have
Thus, by (2.17), we obtain the following corollary.
Corollary 2.3
For \(n \geq0\), we have
The \((h_{1},\ldots,h_{r},q)\) -Euler polynomials are defined by the generating function to be
By (2.18), we have
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
Therefore, by (2.19), we obtain the following theorem.
Theorem 2.4
For \(n \geq0\), we get
By replacing t by \(e^{t}-1\) in (2.17), we have
and
Hence, by (2.20) and (2.21), we obtain the following theorem.
Theorem 2.5
For \(n \geq0\), we have
Let us define the \((h_{1},\ldots,h_{r},q)\) -Boole polynomials of the first kind as follows:
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:
By (2.23), we can see easily
By (2.23) and (2.24), we obtain the following theorem.
Theorem 2.6
For \(n \geq0\), we have
Replacing t as \(e^{t}-1\) in (2.23), we get
and
Hence, by (2.25) and (2.26), we obtain the following theorem.
Theorem 2.7
For \(n \geq0\), we have
From (2.23), we get
Thus, by (2.27), we obtain the following theorem.
Theorem 2.8
For \(n\geq0\), we have
Now, we define the \((h,q)\) -Boole polynomials of the second kind as follows:
By (2.28), we have
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:
By replacing t by \(e^{t}-1\) in (2.30), we have
and
By (2.31) and (2.32), we obtain the following theorem.
Theorem 2.9
For \(n \geq0\), we have
and
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:
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:
Hence, by (2.34), we obtain the following proposition.
Proposition 2.10
For \(n \geq0\), we have
Note that
and, by a similar method, we get
By (2.35) and (2.36), we obtain the following theorem.
Theorem 2.11
For \(n \geq0\), we have
and
By Theorem 2.11, we obtain the following corollary.
Corollary 2.12
For \(n \geq0\), we have
where \(\binom{n}{p,q,r}=\frac{n!}{p!q!r!}\), \(p+q+r=n\).
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The authors are grateful for the valuable comments and suggestions of the referees.
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Kwon, J., Park, JW. A note on \((h,q)\)-Boole polynomials. Adv Differ Equ 2015, 198 (2015). https://doi.org/10.1186/s13662-015-0536-1
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DOI: https://doi.org/10.1186/s13662-015-0536-1
Keywords
- \((h,q)\)-Euler polynomials
- \((h,q)\)-Boole numbers and polynomials
- p-adic invariant integral on \({{\mathbb{Z}}_{p}}\)