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A note on modified degenerate q-Daehee polynomials and numbers

  • Jeong Gon Lee
  • Won Joo Kim
  • Lee-Chae JangEmail author
  • Byung Moon Kim
Open Access
Research
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Abstract

We consider the modified degenerate q-Daehee polynomials and numbers of the second kind which can be represented as the p-adic q-integral. Furthermore, we investigate some properties of those polynomials and numbers.

Keywords

Modified q-Daehee polynomials and numbers Modified degenerate q-Daehee polynomials and numbers 

1 Introduction

Throughout this paper, \(\mathbb{Z}\), \(\mathbb{Q}\), \({\mathbb{Z}}_{p}\), \({\mathbb{Q}}_{p}\) and \({\mathbb{C}}_{p}\) will, respectively, denote the ring of integers, the field of rational numbers, the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of \({\mathbb{Q}}_{p}\). The p-adic norm \(\vert \cdot \vert _{p}\) is normalized by \(\vert p \vert _{p}=\frac{1}{p}\). If \(q \in {\mathbb{C}}_{p}\), we normally assume \(\vert q-1 \vert _{p}< p^{-\frac{1}{p-1}}\), so that \(q^{x} = \exp (x \log q)\) for \(\vert x \vert _{p} \le 1\). The q-extension of x is defined as \([x]_{q}=\frac{1-q^{x}}{1-q}\) for \(q\neq 1\) and x for \(q=1\) (see [3, 4, 5, 6, 12, 17, 18, 20, 21, 25, 27, 29, 30, 31, 33, 34, 35, 41, 45, 46]). Let \(\operatorname{UD}(\mathbb{Z}_{p})\) be the space of uniformly differentiable functions on \(\mathbb{Z}_{p}\). For \(f \in \operatorname{UD}(\mathbb{Z}_{p} )\), Volkenborn integral (or p-adic bosonic integral) on \(\mathbb{Z}_{p}\) is given by
$$ I_{1}(f) = \int _{\mathbb{Z}_{p}} f(x) \,d\mu _{1}(x) = \lim _{N \rightarrow \infty } \frac{1}{p^{N}} \sum_{x=0} ^{p^{N}-1} f(x), $$
(1.1)
where \(\mu _{1}(x)=\mu _{1}(x+p^{N} {\mathbb{Z}}_{p}) \) denotes the Haar distribution defined by \(\mu _{1}(x+p^{N} {\mathbb{Z}}_{p})=\frac{1}{p ^{N}}\) (see [1, 2, 8, 9, 10, 11, 12, 13, 14, 16, 19, 24, 32, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47]). Then, by (1.1), we get \(I(f_{1} ) -I_{1} (f) =f^{\prime } (0)\), where \(f_{1} (x) =f(x+1)\) and \(\frac{d}{dx} f(x)| _{x=0} =f^{\prime } (0)\).
For \(f \in \operatorname{UD}(\mathbb{Z}_{p})\), the p-adic q-integral on \(\mathbb{Z}_{p}\) is defined by Kim to be
$$ I_{q}(f) = \int _{\mathbb{Z}_{p}} f(x) \,d\mu _{q}(x) = \lim _{N \rightarrow \infty } \frac{1}{[p^{N}]_{q}} \sum_{x=0} ^{p^{N}-1} f(x) q^{x} $$
(1.2)
(see [12, 17, 18, 19, 20, 25, 29, 31, 33, 34, 47]). Note that
$$ \lim_{q \rightarrow 1} I_{q}(f) = \lim _{N \rightarrow \infty } \frac{1}{p ^{N}}\sum_{x=0}^{p^{N}-1} f(x) =I_{1}(f) $$
(see [6, 9, 18, 19, 21, 25, 28, 29, 32, 33, 34, 36, 38, 42, 43, 47]). Let \(f_{1} (x) =f(x+1)\). Then, by (1.2), we get
$$ qI_{q} (f_{1} ) -I_{q} (f) =q (q-1) f(0) + \frac{q(q-1)}{\log q} f ^{\prime } (0), $$
(1.3)
where \(f^{\prime } (0) = \frac{d}{dx} f(x)| _{x=0}\) (see [6, 9, 18, 19, 21, 25, 28, 29, 32, 33, 34, 36, 38, 42, 43, 47]).
Carlitz considered q-Bernoulli numbers which are recursively given by
$$ \beta _{0,q}=1,\quad\quad q(q\beta _{q}+1)^{n}- \beta _{n,q} = \textstyle\begin{cases} 1, & \text{if } n=1, \\ 0, & \text{if } n>1, \end{cases} $$
with the usual convention about replacing \(\beta _{q}^{n}\) by \(\beta _{n,q}\) (see [3, 4, 5]). He also defined q-Bernoulli polynomials as
$$ \beta _{n,q}(x)=\sum_{l=0}^{n} \binom{n }{l}[x]_{q}^{n-l}q^{lx} \beta _{l,q},\quad (n \geq 0) \quad (\text{see [3]} ) $$
(see [3, 4, 5]). In [19], Kim proved that the Carlitz q-Bernoulli polynomials are represented by p-adic q-integral on \(\mathbb{Z} _{p}\) as follows:
$$ \int _{\mathbb{Z}_{p}}[x+y]_{q}^{n} \,dm u_{q}(y)=\beta _{n,q} (x) \quad (n\geq 0). $$
(1.4)
In [17], Kim considered the modified q-Bernoulli polynomials which are different from Carlitz to be
$$ B_{n,q}(x)= \int _{\mathbb{Z}_{p}}[x+y]_{q}^{n} \,dm u_{1}(y) \quad (n\geq 0). $$
When \(x=0\), \(B_{n,q}=B_{n,q}(0)\) are called the modified q-Bernoulli numbers (see [17, 18]). Thus, we note that
$$ B_{0,q}=1, \quad\quad (qB_{q}+1)^{n}-B_{n,q} = \textstyle\begin{cases} \frac{\log q}{q-1}, & \text{if } n=1, \\ 0, & \text{if } n>1, \end{cases} $$
with the usual convention about replacing \(B_{q}^{n}\) by \(B_{n,q}\) (see [17, 18, 21, 25, 34]).
In [33, 35, 46], the authors studied the q-Daehee polynomials which are defined by the generating function to be
$$ \int _{\mathbb{Z}_{p}} (1+t)^{x+y} \,d\mu _{q}(y) = \frac{q-1 + \frac{q-1}{ \log q} \log (1+t)}{qt+q-1} (1+t)^{x} =\sum_{n=0}^{\infty } D_{n,q} (x) \frac{t^{n}}{n!} . $$
(1.5)
In [12], the authors studied the degenerate λ-q-Daehee polynomials as follows:
$$\begin{aligned} & \frac{q-1 + \frac{q-1}{\log q} \lambda \log (1+ \frac{1}{u} \log (1+ut) )}{q (1+ \frac{1}{u} \log (1+ut) )^{\lambda } -1} \biggl( 1+\frac{1}{u} \log (1+ut) \biggr)^{x} \\ &\quad = \int _{\mathbb{Z}_{p}} \biggl(1 + \frac{1}{u} \log (1+ut) \biggr)^{\lambda y + x} \,d\mu _{q} (y) \\ &\quad = \sum_{n=0}^{\infty } D_{n, \lambda , q} (x| u) \frac{t^{n}}{n!}. \end{aligned}$$
(1.6)
Like this idea of the Carlitz q-Bernoulli polynomials (1.4), we will define the modified q-Daehee polynomials of the second kind which are different from the modified q-Daehee numbers and polynomials in [31].
As is well known, the Stirling number of the first kind is defined by
$$ (x)_{n} =x (x-1) \cdots (x-n+1) =\sum _{l=0}^{n} S_{1} (n,l) x^{l} , $$
(1.7)
and the Stirling number of the second kind is given by the generating function,
$$ \bigl(e^{t} -1 \bigr)^{m} =m! \sum _{l=m}^{\infty } S_{2} (l,m) \frac{t^{l}}{l!} . $$
(1.8)
We also have
$$ \bigl(\log (1+t) \bigr)^{m} =m! \sum _{n=m}^{\infty } S_{1} (n,m) \frac{t^{n}}{n!} $$
(1.9)
and
$$ x^{n} = \sum_{k=0}^{n} S_{2} (n,k) (x)_{k} $$
(1.10)
(see [7, 14, 15, 22, 23, 26, 28, 48]).

In this paper, we consider the modified q-Daehee polynomials of the second kind and investigate their properties. Furthermore, we consider the modified degenerate q-Daehee polynomials of the second kind and investigate their properties.

2 The modified q-Daehee polynomials and numbers of the second kind

Let p be a fixed prime number. We assume that \(t \in \mathbb{C}_{p}\) with \(\vert t \vert _{p} < p^{-\frac{1}{p-1}}\) and \(q\in \mathbb{C}_{p}\) with \(\vert 1-q \vert _{p}< p^{-\frac{1}{p-1}} \).

The modified q-Daehee polynomials of the second kind are defined by
$$ \int _{\mathbb{Z}_{p}} (1+t)^{[x+y]_{q}} \,d\mu _{0}(y) = \sum_{n=0}^{ \infty } D_{n,q}^{*} (x) \frac{t^{n}}{n!} . $$
(2.1)
When \(x=0\), \(D_{n,q}^{*} =D_{n,q}^{*} (0)\) are called the nth modified q-Daehee numbers of the second kind. By using the binomial theorem in (2.1), we observe that
$$ \int _{\mathbb{Z}_{p}} (1+t)^{[x+y]_{q}} \,d\mu _{0}(y) = \sum_{n=0}^{ \infty } \int _{\mathbb{Z}_{p}} \bigl([x+y]_{q} \bigr)_{n} \,d \mu _{0} (y) \frac{t ^{n}}{n!} . $$
(2.2)
Note that the modified q-Daehee polynomials were defined by Lim in [31] as follows:
$$ D_{n} (x| q)= \int _{\mathbb{Z}_{p}} q^{-y} (x+y)_{n} \,d\mu _{q}(y) . $$
(2.3)
From (2.1) and (2.2), we obtain the following theorem.

Theorem 2.1

For\(n\geq 0\), we have
$$ D_{n,q}^{*} (x) = \int _{\mathbb{Z}_{p}} \bigl([x+y]_{q} \bigr)_{n} \,d \mu _{0} (y) . $$
(2.4)
From (2.1), we derive that
$$ \begin{aligned}[b] \int _{\mathbb{Z}_{p}} (1+t)^{[x+y]_{q}} \,d\mu _{0}(y) ={} & \int _{\mathbb{Z}_{p}} e^{[x+y]_{q} \log (1+t)} \,d\mu _{0}(y) \\ = {}& \sum_{m=0}^{\infty } \int _{\mathbb{Z}_{p}} [x+y]_{q}^{m} \,d\mu _{0}(y) \frac{1}{m!} \bigl(\log (1+t) \bigr)^{m}. \end{aligned} $$
(2.5)
By using (1.9) and (1.10) in Eq. (2.4), we have
$$\begin{aligned} &\sum_{m=0}^{\infty } \int _{\mathbb{Z}_{p}} [x+y]_{q}^{m} \,d\mu _{0}(y) \frac{1}{m!} \bigl(\log (1+t) \bigr)^{m} \\ &\quad = \sum_{m=0}^{\infty } \int _{\mathbb{Z}_{p}} \sum_{k=0}^{m} S_{2} (m,k) \bigl([x+y]_{q} \bigr)_{k} \,d\mu _{0}(y) \sum_{n=m}^{\infty } S_{1} (n,m) \frac{t ^{n}}{n!} \\ &\quad = \sum_{n=0}^{\infty } \Biggl( \sum _{m=0}^{n} \sum_{k=0}^{m} S_{2} (m,k) S_{1} (n,m) \int _{\mathbb{Z}_{p}} \bigl([x+y]_{q} \bigr)_{k} \,d \mu _{0} (y) \Biggr) \frac{t^{n}}{n!} \\ &\quad = \sum_{n=0}^{\infty } \Biggl( \sum _{m=0}^{n} \sum_{k=0}^{m} S_{2} (m,k) S_{1} (n,m) D_{k,q}^{*} (x) \Biggr) \frac{t^{n}}{n!}. \end{aligned}$$
(2.6)

Thus, by (2.1), (2.5), and (2.6), we obtain the following theorem.

Theorem 2.2

For\(n\geq 0\), we have
$$ D_{n,q}^{*} (x) = \sum _{m=0}^{n} \sum_{k=0}^{m} S_{2} (m,k) S_{1} (n,m) D_{k,q}^{*} (x). $$
(2.7)
From (2.1), by replacing t by \(e^{t} -1\) and using (1.8), we get
$$\begin{aligned} \int _{\mathbb{Z}_{p}} e^{[x+y]_{q} t} \,d\mu _{0}(y) = {}& \sum_{m=0}^{ \infty } D_{m,q}^{*} (x) \frac{(e^{t} -1)^{m}}{m!} \\ ={} & \sum_{m=0}^{\infty } D_{m,q}^{*} (x) \sum_{n=m}^{\infty } S_{2} (n,m) \frac{t^{n}}{n!} \\ = {}& \sum_{n=0}^{\infty } \Biggl( \sum _{m=0}^{n} D_{m,q}^{*} (x) S _{2} (n,m) \Biggr) \frac{t^{n}}{n!} , \end{aligned}$$
(2.8)
and by using (1.10) and (2.3), we have
$$\begin{aligned} \int _{\mathbb{Z}_{p}} e^{[x+y]_{q} t} \,d\mu _{0}(y) = {}& \int _{\mathbb{Z}_{p}} \sum_{n=0}^{\infty } [x+y]_{q}^{n} \frac{t^{n}}{n!} \,d\mu _{0} (y) \\ = {}& \sum_{n=0}^{\infty } \int _{\mathbb{Z}_{p}} [x+y]_{q}^{n} \,d\mu _{0} (y) \frac{t^{n}}{n!} \\ = {}& \sum_{n=0}^{\infty } \int _{\mathbb{Z}_{p}} \bigl( [x]_{q} +q ^{x} [y]_{q} \bigr)^{n} \,d\mu _{0} (y) \frac{t^{n}}{n!} \\ = {}& \sum_{n=0}^{\infty } \Biggl( \sum _{k=0}^{n} \binom{n}{k} [x]_{q} ^{n-k} q^{kx} \int _{\mathbb{Z}_{p}} [y]_{q}^{k} \,d\mu _{0} (y) \Biggr) \frac{t^{n}}{n!} \\ = {}& \sum_{n=0}^{\infty } \Biggl( \sum _{k=0}^{n} \binom{n}{k} [x]_{q} ^{n-k} q^{kx} \int _{\mathbb{Z}_{p}} \sum_{l=0}^{k} S_{2}(k,l) \bigl([y]_{q} \bigr)_{l} \,d\mu _{0} (y) \Biggr) \frac{t^{n}}{n!} \\ = {}&\sum_{n=0}^{\infty } \Biggl( \sum _{k=0}^{n} \sum_{l=0}^{k} \binom{n}{k} [x]_{q}^{n-k} q^{kx} S_{2}(k,l) D_{l,q}^{*} \Biggr) \frac{t ^{n}}{n!}. \end{aligned}$$
(2.9)
From (2.8) and (2.9), we obtain the following theorem.

Theorem 2.3

For\(n\geq 0\), we have
$$ \sum_{m=0}^{n} D_{m,q}^{*} (x) S_{2} (n,m) = \sum _{k=0}^{n} \sum_{l=0} ^{k} \binom{n}{k} [x]_{q}^{n-k} q^{kx} S_{2}(k,l) D_{l,q}^{*}. $$
(2.10)

3 The modified degenerate q-Daehee polynomials of the second kind

Let p be a fixed prime number. We assume that \(t \in \mathbb{C}_{p}\) with \(\vert t \vert _{p} < p^{-\frac{1}{p-1}}\).

The modified degenerate q-Daehee polynomials of the second kind are defined by
$$ \int _{\mathbb{Z}_{p}} \biggl( 1+ \frac{1}{\lambda } \log (1+\lambda t) \biggr)^{[x+y]_{q}} \,d\mu _{0}(y)=\sum _{n=0}^{\infty } D_{n,\lambda ,q} ^{*} (x) \frac{t^{n}}{n!} . $$
(3.1)
When \(x=0\), \(D_{n, \lambda , q}^{*} =D_{n, \lambda ,q}^{*} (0)\) are called the modified degenerate q-Daehee numbers of the second kind.
We note that the reason for calling \(D_{n, \lambda , q}^{*}\) the modified degenerate q-Daehee polynomials of the second kind is to distinguish it from the modified q-Daehee numbers and polynomials in [31]. From (3.1), we observe that
$$\begin{aligned} \int _{\mathbb{Z}_{p}} \biggl(1+ \frac{1}{\lambda } \log (1+\lambda t) \biggr)^{[x+y]_{q}} \,d\mu _{0}(y) = {}& \sum _{m=0}^{\infty } \int _{\mathbb{Z}_{p}} \binom{[x+y]_{q}}{m} \,d\mu _{0} (y) \biggl( \frac{1}{ \lambda } \log (1+\lambda t) \biggr)^{m} \\ = {}& \sum_{m=0}^{\infty } \int _{\mathbb{Z}_{p}} \bigl([x+y]_{q} \bigr)_{m} \,d \mu _{0} (y) {\lambda }^{-m} \frac{1}{m!} \bigl(\log (1+\lambda t) \bigr)^{m} \\ = {}& \sum_{m=0}^{\infty } \bigl(D_{m,q}^{*} (x) {\lambda }^{-m} \bigr) \Biggl(\sum_{n=m}^{\infty } {\lambda }^{n} S_{1} (n,m) \frac{t^{n}}{n!} \Biggr) \\ ={} & \sum_{n=0}^{\infty } \Biggl( \sum _{m=0}^{n} D_{m,q}^{*} (x) { \lambda }^{n-m} S_{1} (n,m) \Biggr) \frac{t^{n}}{n!} . \end{aligned}$$
(3.2)
From (3.1) and (3.2), we obtain the following theorem.

Theorem 3.1

For\(n\geq 0\), we have
$$ D_{n,\lambda , q}^{*} (x) =\sum _{m=0}^{n} D_{m,q}^{*} (x) \lambda ^{n-m} S_{1} (n,m). $$
(3.3)
From (3.1), by replacing t by \(\frac{1}{\lambda } (e^{\lambda t} -1)\), we derive
$$\begin{aligned} \int _{\mathbb{Z}_{p}} (1+ t)^{[x+y]_{q}} \,d\mu _{0}(y) = {}& \sum_{m=0} ^{\infty } D_{m,\lambda ,q}^{*} (x) \frac{(\frac{1}{\lambda } (e^{ \lambda t} -1))^{m}}{m!} \\ ={} & \sum_{m=0}^{\infty } D_{m,\lambda ,q}^{*}(x) {\lambda }^{-m} \sum_{n=m}^{\infty } S_{2} (n,m) \frac{\lambda ^{n} t^{n}}{n!} \\ ={} & \sum_{n=0}^{\infty } \sum _{m=0}^{n} D_{m,\lambda ,q}^{*} (x) { \lambda }^{n-m} S_{2} (n,m) \frac{t^{n}}{n!} . \end{aligned}$$
(3.4)
From (3.4) and (2.1), we obtain the following theorem.

Theorem 3.2

For\(n\geq 0\), we have
$$ D_{n,q}^{*} (x) =\sum _{m=0}^{n} D_{m,\lambda ,q}^{*} (x)\lambda ^{n-m} S_{2} (n,m). $$
(3.5)
From (3.1), we observe that
$$\begin{aligned} \biggl( 1+ \frac{1}{\lambda } \log (1+\lambda t) \biggr)^{[x+y]_{q}} = {}& e^{[x+y]_{q} \log (1+ \frac{1}{\lambda } \log (1+\lambda t))} \\ ={} & \sum_{m=0}^{\infty } [x+y]_{q}^{m} \biggl(\log \biggl(1+ \frac{1}{ \lambda } \log (1+\lambda t) \biggr) \biggr)^{m} \frac{1}{m!} \\ ={} & \sum_{m=0}^{\infty } [x+y]_{q}^{m} \sum_{l=m}^{\infty } S_{1} (l,m) \frac{(\frac{1}{\lambda } \log (1+\lambda t))^{l}}{l!} \\ ={} & \sum_{l=0}^{\infty } \sum _{m=0}^{l} [x+y]_{q}^{m} S_{1} (l,m) {\lambda }^{-l} \sum _{n=l}^{\infty } S_{1} (n,l) {\lambda }^{n} \frac{t ^{n}}{n!} \\ ={} & \sum_{n=0}^{\infty } \Biggl(\sum _{l=0}^{n} \sum_{m=0}^{l} [x+y]_{q} ^{m} S_{1} (l,m) {\lambda }^{n-l} S_{1} (n,l) \Biggr) \frac{t^{n}}{n!}. \end{aligned}$$
(3.6)
From (3.7), we get
$$\begin{aligned} & \int _{\mathbb{Z}_{p}} \biggl( 1+ \frac{1}{\lambda } \log (1+\lambda t) \biggr) ^{[x+y]_{q}} \,d\mu _{0}(y) \\ &\quad = \sum_{n=0}^{\infty } \Biggl(\sum _{l=0}^{n} \sum_{m=0}^{l} \sum_{k=0}^{m} S_{2} (m,k) S_{1} (l,m) \lambda ^{n-l} S_{1} (n,l) \int _{\mathbb{Z}_{p}} \bigl([x+y]_{q} \bigr)_{k} \,d \mu _{0} (y) \Biggr) \frac{t^{n}}{n!} \\ &\quad = \sum_{n=0}^{\infty } \Biggl( \sum _{l=0}^{n} \sum_{m=0}^{l} \sum_{k=0}^{m} \lambda ^{n-l} S_{1} (l,m) S_{1} (n,l) S_{2} (m,k) D_{k,q} ^{*} (x) \Biggr) \frac{t^{n}}{n!} . \end{aligned}$$
(3.7)
From (3.7) and (3.1), we obtain the following theorem.

Theorem 3.3

For\(n\geq 0\), we have
$$ D_{n,\lambda ,q}^{*} (x) =\sum _{l=0}^{n} \sum_{m=0}^{l} \sum_{k=0} ^{m} \lambda ^{n-l} S_{1} (l,m) S_{1} (n,l) S_{2} (m,k) D_{k,q}^{*} (x). $$
(3.8)

4 Conclusion

Many authors studied the q-Daehee polynomials (1.5), the degenerate λ-q-Daehee polynomials of the second kind in [12, 33, 46]. In this paper, we defined the modified q-Daehee polynomials of the second kind (2.1), which are different from the q-Daehee polynomials (1.5), and the modified degenerate q-Daehee polynomials of the second kind (3.1), which are different from the modified q-Daehee numbers and polynomials in [31]. We obtained the interesting results of Theorems 2.1, 2.2, and 2.3, which are some identity properties related with the modified degenerate q-Daehee polynomials of the second kind (3.1) and also we obtained the results of Theorems 3.1, 3.2, and 3.3, which are some identities related with the modified q-Daehee polynomials of the second kind.

Notes

Authors’ contributions

All authors contributed equally to this work. All authors read and approved the final manuscript.

Funding

This paper was supported by Wonkwang University in 2017.

Competing interests

The authors declare that they have no competing interests.

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© The Author(s) 2019

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 and Affiliations

  • Jeong Gon Lee
    • 1
  • Won Joo Kim
    • 2
  • Lee-Chae Jang
    • 3
    Email author
  • Byung Moon Kim
    • 4
  1. 1.Division of Applied Mathematics, Nanoscale Science and Technology InstituteWonkwang UniversityIksanRepublic of Korea
  2. 2.Department of Applied MathematicsKyunghee UniversitySeoulRepublic of Korea
  3. 3.Graduate School of EducationKonkuk UniversitySeoulRepublic of Korea
  4. 4.Department of Mechanical System EngineeringDongguk UniversityGyeongjuRepublic of Korea

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