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On \((h,q)\)-Daehee numbers and polynomials

Advances in Difference Equations20152015:107

https://doi.org/10.1186/s13662-015-0445-3

Received: 10 February 2015

Accepted: 17 March 2015

Published: 31 March 2015

Abstract

The p-adic q-integral (sometimes called q-Volkenborn integration) was defined by Kim. From p-adic q-integral equations, we can derive various q-extensions of Bernoulli polynomials and numbers. DS Kim and T Kim studied Daehee polynomials and numbers and their applications. Kim et al. introduced the q-analogue of Daehee numbers and polynomials which are called q-Daehee numbers and polynomials. Lim considered the modified q-Daehee numbers and polynomials which are different from the q-Daehee numbers and polynomials of Kim et al. In this paper, we consider \((h,q)\)-Daehee numbers and polynomials and give some interesting identities. In case \(h=0\), we cover the q-analogue of Daehee numbers and polynomials of Kim et al. In case \(h=1\), we modify q-Daehee numbers and polynomials. We can find out various \((h,q)\)-related numbers and polynomials which are studied by many authors.

Keywords

\((h,q)\)-Daehee numbers \((h,q)\)-Daehee polynomials \((h,q)\)-Bernoulli polynomials p-adic q-integral

MSC

11B6811S40

1 Introduction

Let p be a fixed 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 completion of algebraic closure of \(\mathbb{Q}_{p}\). The p-adic norm is defined \(|p|_{p}=\frac{1}{p}\).

When one talks of q-extension, q is variously considered as an indeterminate, 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 \(\lim_{q\rightarrow1}[x]_{q}=x\) for each \(x\in\mathbb{Z}_{p}\).
Let \(UD(\mathbb{Z}_{p})\) be the space of a uniformly differentiable function on \(\mathbb{Z}_{p}\). For \(f\in UD(\mathbb{Z}_{p})\), the p-adic q-integral on \(\mathbb{Z}_{p}\) is defined by Kim as follows:
$$ I_{q}(f)=\int_{\mathbb{Z}_{p}}f(x)\,d\mu_{q}(x)=\lim_{N\rightarrow\infty }\frac{1}{[p^{N}]_{q}}\sum _{x=0}^{p^{N}-1}f(x)q^{x} \quad(\mbox{see [1, 2]}). $$
(1)
Using this integration, the q-Daehee polynomials \(D_{n,q}(x)\) are defined and studied by Kim et al. (see [3]), their generating function is as follows:
$$ \frac{1-q+\frac{1-q}{\log q}\log(1+t)}{1-q-qt}(1+t)^{x}=\sum _{n=0}^{\infty}D_{n,q}(x)\frac{t^{n}}{n!}. $$
(2)
The generating function of the modified q-Daehee polynomials are defined and studied by Lim (see [4]).
$$ F_{q}(x,t)=\frac{q-1}{\log q}\frac{\log(1+t)}{t}(1+t)^{x}= \sum_{n=0}^{\infty}D_{n}(x|q) \frac{t^{n}}{n!} \quad(\mbox{see [1--16]}). $$
(3)
From (1), we have the following integral identity:
$$ qI_{q}(f_{1})-I_{q}(f)= \frac{q-1}{\log q}f'(0)+(q-1)f(0), $$
(4)
where \(f_{1}(x)=f(x+1)\) and \(\frac{d}{dx}f(x)=f'(x)\).
In a special case, for \(h\in\mathbb{Z}_{+}\) (\(=\mathbb{N}\cup\{0\}\)), we apply \(f(x)=q^{-hx}e^{tx}\) on (4), we have
$$ \int_{\mathbb{Z}_{p}}q^{-hx}e^{xt}\,d\mu_{q}(x)=\frac{q^{h-1}(q-1)}{\log q}\frac{t-(h-1)\log{q}}{e^{t}-q^{h-1}}. $$
(5)
For \(h\in\mathbb{Z}_{+}\), we define the \((h,q)\)-Bernoulli number \(B^{(h)}_{n}(q)\) as follows:
$$ \sum_{n=0}^{\infty}B^{(h)}_{n}(q) \frac{t^{n}}{n!}=\frac {q^{h-1}(q-1)}{\log q}\frac{t-(h-1)\log{q}}{e^{t}-q^{h-1}}. $$
(6)
Indeed if \(q\rightarrow1\), we have \(\lim_{q\rightarrow 1}B^{(h)}_{n}(q)=B_{n}\). So we call this \(B^{(h)}_{n}(q)\) the nth \((h,q)\)-Bernoulli number. And we define \((h,q)\)-Bernoulli polynomials and the generating function to be
$$ \frac{q^{h-1}(q-1)}{\log q}\frac{t-(h-1)\log{q}}{e^{t}-q^{h-1}}e^{xt}=\sum _{n=0}^{\infty}B^{(h)}_{n}(x|q) \frac{t^{n}}{n!}. $$
(7)

When \(x=0\), \(B^{(h)}_{n}(0|q)=B^{(h)}_{n}(q)\) are the nth \((h,q)\)-Bernoulli numbers.

From (4) and (7), we have
$$ B^{(h)}_{n}(x|q)=\int_{\mathbb{Z}_{p}}q^{-hy}(x+y)^{n}\,d \mu_{q}(y). $$
From (7) we note that
$$ B^{(h)}_{n}(x|q)=\sum _{l=0}^{n}\binom{n}{l}B^{(h)}_{l}(q)x^{n-l}. $$
(8)
For the case \(|t|_{p}\leq p^{-\frac{1}{p-1}}\), the Daehee polynomials are defined as follows (see [3]):
$$ \sum_{n=0}^{\infty}D_{n}(x) \frac{t^{n}}{n!}=\frac{\log(1+t)}{t}(1+t)^{x}. $$
(9)
From (2) and (3), if \(q\rightarrow1\), we have
$$ \lim_{q\rightarrow1}D_{n,q}(x)=D_{n}(x) $$
and
$$ \lim_{q\rightarrow1}D_{n}(x|q)=D_{n}(x). $$

The p-adic q-integral (or q-Volkenborn integration) was defined by Kim (see [1, 2]). From p-adic q-integral equations, we can derive various q-extensions of Bernoulli polynomials and numbers (see [124]). In [20], DS Kim and T Kim studied Daehee polynomials and numbers and their applications. In [3], Kim et al. introduced the q-analogue of Daehee numbers and polynomials which are called q-Daehee numbers and polynomials. Lim considered in [4] the modified q-Daehee numbers and polynomials which are different from the q-Daehee numbers and polynomials of Kim et al. In this paper, we consider \((h,q)\)-Daehee numbers and polynomials and give some interesting identities. In case \(h=0\), we cover the q-analogue of Daehee numbers and polynomials of Kim et al. (see [3]). In case \(h=1\), we have modified q-Daehee numbers and polynomials in [4]. We can find out various \((h,q)\)-related numbers and polynomials in [10, 13, 14].

2 \((h,q)\)-Daehee numbers and polynomials

Let us now consider the p-adic q-integral representation as follows: for each \(h\in\mathbb{Z}_{+}\),
$$ \int_{\mathbb{Z}_{p}}q^{-hy}(x+y)_{n}\,d \mu_{q}(y)\quad \bigl(n\in\mathbb{Z}_{+}=\mathbb {N}\cup\{0\} \bigr), $$
(10)
where \((x)_{n}\) is known as the Pochhammer symbol (or decreasing factorial) defined by
$$ (x)_{n}=x(x-1)\cdots(x-n+1)=\sum _{k=0}^{n}S_{1}(n,k)x^{k}, $$
(11)
and here \(S_{1}(n,k)\) is the Stirling number of the first kind (see [3, 20]).
From (10) we have
$$ \begin{aligned}[b] \sum_{n=0}^{\infty} \biggl(\int_{\mathbb{Z}_{p}}q^{-hy}(y)_{n}\,d\mu _{q}(y) \biggr)\frac{t^{n}}{n!}&=\int_{\mathbb{Z}_{p}}q^{-hy} \Biggl(\sum_{n=0}^{\infty}\binom{y}{n}t^{n} \Biggr)\,d\mu_{q}(y) \\ &=\int_{\mathbb{Z}_{p}}q^{-hy}(1+t)^{y}\,d\mu_{q}(y), \end{aligned} $$
(12)
where \(t\in\mathbb{C}_{p}\) with \(|t|_{p}< p^{-\frac{1}{p-1}}\).
For \(|t|_{p}< p^{-\frac{1}{p-1}}\), from (4) we have
$$ \int_{\mathbb{Z}_{p}}q^{-hy}(1+t)^{y}\,d\mu_{q}(y)=\frac{q^{h-1}(q-1)}{\log q}\frac{\log{\frac{1+t}{q^{h-1}}}}{1+t-q^{h-1}}. $$
(13)
Let
$$ F^{(h)}_{q}(t)=\frac{q^{h-1}(q-1)}{\log q} \frac{\log{\frac {1+t}{q^{h-1}}}}{1+t-q^{h-1}}=\sum_{n=0}^{\infty }D^{(h)}_{n}(q) \frac{t^{n}}{n!}. $$
(14)
Here, the numbers \(D^{(h)}_{n}(q)\) are called the nth \((h,q)\)-Daehee numbers of the first kind. Moreover, we have
$$ D^{(h)}_{n}(q)=\int_{\mathbb{Z}_{p}}q^{-hy}(y)_{n}\,d \mu_{q}(y). $$
(15)

From (14) and (15), if \(h=0\), \(D^{(0)}_{n}(q)\) is just the q-Daehee numbers which are defined by Kim et al. in [3]. If \(h=1\), \(D^{(1)}_{n}(q)\) is just the modified q-Daehee numbers which are studied in [4].

On the other hand, we can derive \((h,q)\)-Daehee polynomials
$$ \begin{aligned}[b] \sum_{n=0}^{\infty} \biggl(\int_{\mathbb{Z}_{p}}q^{-hy}(x+y)_{n}\,d\mu _{q}(y) \biggr)\frac{t^{n}}{n!}&=\int_{\mathbb{Z}_{p}}q^{-hy} \Biggl(\sum_{n=0}^{\infty}\binom{x+y}{n}t^{n} \Biggr)\,d\mu_{q}(y) \\ &=\int_{\mathbb{Z}_{p}}q^{-hy}(1+t)^{x+y}\,d\mu_{q}(y) \\ &=\frac{q^{h-1}(q-1)}{\log q}\frac{\log{(1+t)}-(h-1)\log {q}}{1+t-q^{h-1}}(1+t)^{x} \\ &=\sum_{n=0}^{\infty}D^{(h)}_{n}(x|q) \frac{t^{n}}{n!}, \end{aligned} $$
(16)
where \(t\in\mathbb{C}_{p}\) with \(|t|_{p}< p^{-\frac{1}{p-1}}\).

When \(x=0\), \(D^{(h)}_{n}(0|q)=D^{(h)}_{n}(q)\) is called the nth \((h,q)\)-Daehee number.

Notice that \(F^{(h)}_{q}(0,t)\) seems to be a new q-extension of the generating function for Daehee numbers of the first kind. Therefore, from (9) and the following fact, we get
$$ \lim_{q\rightarrow1}F^{(h)}_{q}(t)= \frac{\log(1+t)}{t}. $$
From (11) and (12), we have
$$ D^{(h)}_{n}(x|q)=\int _{\mathbb{Z}_{p}}q^{-hy}(x+y)_{n}\,d\mu_{q}(y) =\sum_{k=0}^{n}S_{1}(n,k)B^{(h)}_{k}(x|q), $$
(17)
where \(B^{(h)}_{k}(x|q)\) are the \((h,q)\)-Bernoulli polynomials introduced in (7).

Thus we have the following theorem, which relates \((h,q)\)-Bernoulli polynomials and \((h,q)\)-Daehee polynomials.

Theorem 1

For \(n,m\in\mathbb{Z}_{+}\), we have the following equalities:
$$ D^{(h)}_{n}(x|q)=\sum_{k=0}^{n}S_{1}(n,k)B^{(h)}_{k}(x|q) $$
and
$$ D^{(h)}_{n}(q)=\sum_{k=0}^{n}S_{1}(n,k)B^{(h)}_{k}(q). $$
From the generating function of the \((h,q)\)-Daehee polynomials in \(D^{(h)}_{n}(x|q)\) in (14), by replacing t to \(e^{t}-1\), we have
$$ \begin{aligned}[b] \sum_{n=0}^{\infty}D^{(h)}_{n}(x|q) \frac{(e^{t}-1)^{n}}{n!}&=\frac {q^{h-1}(q-1)}{\log q}\frac{t-(h-1)\log{q}}{e^{t}-q^{h-1}}e^{xt} \\ &=\sum_{n=0}^{\infty}B^{(h)}_{n}(x|q) \frac{t^{n}}{n!}. \end{aligned} $$
(18)
On the other hand,
$$ \sum_{n=0}^{\infty}D^{(h)}_{n}(x|q) \frac{(e^{t}-1)^{n}}{n!}=\sum_{m=0}^{\infty}D^{(h)}_{m}(x|q) \sum_{n=0}^{\infty}S_{2}(n,m) \frac{t^{n}}{n!}. $$
(19)
Here, \(S_{2}(n,m)\) is the Stirling number of the second kind defined by the following generating series:
$$ \sum_{n=m}^{\infty}S_{2}(n,m) \frac{t^{n}}{n!}=\frac {(e^{t}-1)^{m}}{m!} \quad\textit{cf. }\mbox{[3, 20]}. $$
(20)
Thus by comparing the coefficients of \(t^{n}\), we have
$$ B^{(h)}_{n}(x|q)=\sum_{m=0}^{n}D^{(h)}_{m}(x|q)S_{2}(n,m). $$

Therefore, we obtain the following theorem.

Theorem 2

For \(n,m\in\mathbb{Z}_{+}\), we have the following identity:
$$ B^{(h)}_{n}(x|q)=\sum_{m=0}^{n}D^{(h)}_{m}(x|q)S_{2}(n,m). $$
The increasing factorial sequence is known as
$$ x^{(n)}=x(x+1) (x+2)\cdots(x+n-1)\quad (n\in\mathbb{Z}_{+}). $$
Let us define the \((h,q)\)-Daehee numbers of the second kind as follows:
$$ \widehat{D}^{(h)}_{n}(q)=\int _{\mathbb{Z}_{p}}q^{-hy}(-y)_{n}\,d\mu_{q}(y) \quad (n\in\mathbb{Z}_{+}). $$
(21)
It is easy to observe that
$$ x^{(n)}=(-1)^{n}(-x)_{n}=\sum _{k=0}^{n}S_{1}(n,k) (-1)^{n-k}x^{k}. $$
(22)
From (21) and (22), we have
$$ \begin{aligned}[b] \widehat{D}^{(h)}_{n}(q)&=\int _{\mathbb{Z}_{p}}q^{-hy}(-y)_{n}\,d\mu_{q}(y) \\ &=\int_{\mathbb{Z}_{p}}q^{-hy}y^{(n)}(-1)^{n}\,d \mu_{q}(y) \\ &=\sum_{k=0}^{n}S_{1}(n,k) (-1)^{k}B^{(h)}_{k}(q). \end{aligned} $$
(23)

Thus, we state the following theorem, which relates \((h,q)\)-Daehee numbers and \((h,q)\)-Bernoulli numbers.

Theorem 3

The following holds true:
$$ \widehat{D}^{(h)}_{n}(q)=\sum_{k=0}^{n}S_{1}(n,k) (-1)^{k}B^{(h)}_{k}(q). $$
Let us now consider the generating function of \((h,q)\)-Daehee numbers of the second kind as follows:
$$ \begin{aligned}[b] \sum_{n=0}^{\infty} \widehat{D}^{(h)}_{n}(q)\frac{t^{n}}{n!}&=\sum _{n=0}^{\infty} \biggl(\int_{\mathbb{Z}_{p}}q^{-hy}(-y)_{n}\,d \mu _{q}(y) \biggr)\frac{t^{n}}{n!} \\ &=\int_{\mathbb{Z}_{p}}q^{-hy} \Biggl(\sum _{n=0}^{\infty}\binom {-y}{n}t^{n} \Biggr)\,d\mu_{q}(y) \\ &=\int_{\mathbb{Z}_{p}}q^{-hy}(1+t)^{-y}\,d\mu_{q}(y). \end{aligned} $$
(24)
From (4) and (24), we have the generating function for \((h,q)\)-Daehee numbers of the second kind as follows:
$$ \int_{\mathbb{Z}_{p}}q^{-hy}(1+t)^{-y}\,d\mu_{q}(y)=\frac{q^{h-1}(q-1)}{\log q}\frac{\log q-\log(1+t)}{1+t-q^{h-1}}. $$
(25)
Let us consider the \((h,q)\)-Daehee polynomials of the second kind as follows:
$$ \begin{aligned}[b] \sum_{n=0}^{\infty} \widehat{D}^{(h)}_{n}(x|q)\frac{t^{n}}{n!}&=\sum _{n=0}^{\infty}\int_{\mathbb{Z}_{p}}q^{-hy}(x-y)_{n}\,d \mu_{q}(y)\frac {t^{n}}{n!} \\ &=\int_{\mathbb{Z}_{p}}q^{-hy}(1+t)^{x-y}\,d\mu_{q}(y) \\ &=\frac{q^{h-1}(q-1)}{\log q}\frac{\log q-\log(1+t)}{1+t-q^{h}}(1+t)^{x}. \end{aligned} $$
(26)
From the \((h,q)\)-Bernoulli polynomials in (7),
$$ \begin{aligned}[b] q^{h}\sum_{n=0}^{\infty}(-1)^{n}B^{(h)}_{n} \bigl(x|q^{-1} \bigr)\frac {t^{n}}{n!}&=q^{h}\frac{q^{1-h}(q^{-1}-1)}{\log q^{-1}} \frac{-t-\log {q^{1-h}}}{e^{-t}-q^{1-h}}e^{-xt} \\ &=\frac{q^{h-1}(q-1)}{\log q}\frac{t-\log {q^{h-1}}}{e^{t}-q^{h-1}}e^{(1-x)t} \\ &=\sum_{n=0}^{\infty}B^{(h)}_{n}(1-x|q) \frac{t^{n}}{n!}. \end{aligned} $$
(27)
Thus, we have
$$ q^{h}(-1)^{n}B^{(h)}_{n} \bigl(x|q^{-1} \bigr)=B^{(h)}_{n}(1-x|q). $$
(28)
From (28), the value at \(x=1\), we have
$$ q^{h}(-1)^{n}B^{(h)}_{n} \bigl(1|q^{-1} \bigr)=B^{(h)}_{n}(q). $$
On the other hand, we note that
$$ \begin{aligned}[b] (-x)_{n}&=(-1)^{n}x^{(n)} =\sum_{l=0}^{n}S_{1}(n,l) (-x)^{l} =(-1)^{n}\sum_{l=0}^{n}\bigl|S_{1}(n,l)\bigr|x^{l}, \end{aligned} $$
(29)
where \(n\geq0\) and \(|S_{1}(n,k)|\) is the unsigned Stirling number of the first kind.
From (28) and (29),
$$ \begin{aligned}[b] \widehat{D}^{(h)}_{n}(x|q)&= \sum_{l=0}^{n}\bigl|S_{1}(n,l)\bigr|(-1)^{l} \int_{\mathbb{Z}_{p}}q^{-hy}(-x+y)^{l}\,d \mu_{q}(y) \\ &=\sum_{l=0}^{n}\bigl|S_{1}(n,l)\bigr|(-1)^{l}B^{(h)}_{l}(-x|q) \\ &=q^{-h}\sum_{l=0}^{n}\bigl|S_{1}(n,l)\bigr|B^{(h)}_{l} \bigl(x+1|q^{-1} \bigr). \end{aligned} $$
(30)

Thus, we have the following identity.

Theorem 4

For \(n\in\mathbb{Z}_{+}\), the following is true:
$$ \widehat{D}^{(h)}_{n}(x|q)=q^{-h}\sum _{l=0}^{n}\bigl|S_{1}(n,l)\bigr|B^{(h)}_{l} \bigl(x+1|q^{-1} \bigr). $$
On the other hand, we can check easily the following:
$$ (x+y)_{n}=(-1)^{n}(-x-y+n-1)_{n} $$
(31)
and
$$ \frac{(x+y)_{n}}{n!}=(-1)^{n}\binom{-x+y+n-1}{n}. $$
(32)
From (14), (26), (31) and (32), we have
$$ \begin{aligned}[b] (-1)^{n}\frac{D^{(h)}_{n}(x|q)}{n!}&=\int _{\mathbb{Z}_{p}}q^{-hy}\binom {-x-y+n-1}{n}\,d\mu_{q}(y) \\ &=\sum_{m=0}^{n}\binom{n-1}{n-m}\int _{\mathbb{Z}_{p}}q^{-hy}\binom {-x-y}{m}\,d\mu_{q}(y) \\ &=\sum_{m=1}^{n}\binom{n-1}{m-1} \frac{\widehat{D}^{(h)}_{m}(-x|q)}{m!} \end{aligned} $$
(33)
and
$$ \begin{aligned}[b] (-1)^{n}\frac{\widehat{D}^{(h)}_{n}(x|q)}{n!}&=(-1)^{n} \int_{\mathbb {Z}_{p}}q^{-hy}\binom{-x+y}{n}\,d\mu_{q}(y) \\ &=\int_{\mathbb{Z}_{p}}q^{-hy}\binom{-x+y+n-1}{n}\,d\mu_{q}(y) \\ &=\sum_{m=0}^{n}\binom{n-1}{n-m}\int _{\mathbb{Z}_{p}}q^{-hy}\binom {-x+y}{m}\,d\mu_{q}(y) \\ &=\sum_{m=1}^{n}\binom{n-1}{m-1} \frac{D^{(h)}_{m}(-x|q)}{m!}. \end{aligned} $$
(34)

Therefore, we get the following theorem, which relates \((h,q)\)-Daehee polynomials of the first and the second kind.

Theorem 5

For \(n\in\mathbb{N}\), the following equalities hold true:
$$ (-1)^{n}\frac{D^{(h)}_{n}(x|q)}{n!}=\sum_{m=1}^{n} \binom {n-1}{m-1}\frac{\widehat{D}^{(h)}_{m}(-x|q)}{m!} $$
and
$$ (-1)^{n}\frac{\widehat{D}^{(h)}_{n}(x|q)}{n!}=\sum_{m=1}^{n} \binom {n-1}{m-1}\frac{D^{(h)}_{m}(-x|q)}{m!}. $$

Declarations

Acknowledgements

Authors wish to express their sincere gratitude to the referees for their valuable suggestions and comments.

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.

Authors’ Affiliations

(1)
Department of Mathematics, Kyungpook National University, Daegu, S. Korea

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Copyright

© Do and Lim; licensee Springer. 2015

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