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A note on modified degenerate q-Daehee polynomials and numbers

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.

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.

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)

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)

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.

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This paper was supported by Wonkwang University in 2017.

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Lee, J.G., Kim, W.J., Jang, LC. et al. A note on modified degenerate q-Daehee polynomials and numbers. J Inequal Appl 2019, 24 (2019). https://doi.org/10.1186/s13660-019-1966-1

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Keywords

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