# 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

$$\beta _{n,q}(x)=\sum_{l=0}^{n} \binom{n }{l}[x]_{q}^{n-l}q^{lx} \beta _{l,q},\quad (n \geq 0) \quad (\text{see } )$$

(see [3,4,5]). In , 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 , 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 , 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 .

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  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 . 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 . 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.

## References

1. Araci, S., Acikgoz, M.: A note on the Frobenius–Euler numbers and polynomials associated with Bernstein polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 22(3), 399–406 (2012)

2. Bayad, A., Chikhi, J.: Apostol–Euler polynomials and asymptotics for negative binomial reciprocals. Adv. Stud. Contemp. Math. (Kyungshang) 24(1), 33–37 (2014)

3. Carlitz, L.: q-Bernoulli and Eulerian numbers. Trans. Am. Math. Soc. 76, 332–350 (1954)

4. Carlitz, L.: q-Bernoulli numbers and polynomials. Duke Math. J. 25, 987–1000 (1958)

5. Carlitz, L.: Expansions of q-Bernoulli numbers. Duke Math. J. 25, 355–364 (1958)

6. Dolgy, D.V., Jang, G.-W., Kwon, H.-I., Kim, T.: A note on Carlitz’s type q-Changhee numbers and polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 27(4), 451–459 (2017)

7. Dolgy, D.V., Kim, T.: Some explicit formulas of degenerate Stirling numbers associated with the degenerate special numbers and polynomials. Proc. Jangjeon Math. Soc. 21(2), 309–317 (2018)

8. El-Desouky, B.S., Mustafa, A.: New results on higher-order Daehee and Bernoulli numbers and polynomials. Adv. Differ. Equ. 2016, 32 (2016)

9. Jang, G.-W., Kim, T.: Revisit of identities for Daehee numbers arising from nonlinear differential equations. Proc. Jangjeon Math. Soc. 20(2), 163–177 (2017)

10. Jang, G.W., Kim, D.S., Kim, T.: Degenerate Changhee numbers and polynomials of the second kind. Adv. Stud. Contemp. Math. (Kyungshang) 27(4), 609–624 (2017)

11. Khan, W.A., Nisar, K.S., Duran, U., Acikgoz, M., Araci, S.: Multifarious implicit summation formulae of Hermite-based poly-Daehee polynomials. Proc. Jangjeon Math. Soc. 21(3), 305–310 (2018)

12. Kim, B.M., Yun, S.J., Park, J.-W.: On a degenerate λ-q-Daehee polynomials. J. Nonlinear Sci. Appl. 9, 4607–4616 (2016)

13. Kim, D.S., Kim, T.: A note on degenerate Eulerian numbers and polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 27(4), 431–440 (2017)

14. Kim, D.S., Kim, T.: A new approach to Catalan numbers using differential equations. Russ. J. Math. Phys. 24(4), 465–475 (2018)

15. Kim, D.S., Kim, T.: Some p-adic integrals on $$\mathbb{Z}_{p}$$ associated with trigonometric functions. Russ. J. Math. Phys. 25(3), 300–308 (2018)

16. Kim, D.S., Kim, T., Kwon, H.-I., Jang, G.-W.: Degenerate Daehee polynomials of the second kind. Proc. Jangjeon Math. Soc. 21(1), 83–97 (2018)

17. Kim, T.: On explicit formulas of p-adic $$q-L$$-functions. Kyushu J. Math. 48(1), 73–86 (1994)

18. Kim, T.: On p-adic q-Bernoulli numbers. J. Korean Math. Soc. 37(1), 21–30 (2000)

19. Kim, T.: q-Volkenborn integration. Russ. J. Math. Phys. 9(3), 288–299 (2002)

20. Kim, T.: An invariant p-adic q-integral on $$\mathbb{Z}_{p}$$. Appl. Math. Lett. 21(2), 105–108 (2008)

21. Kim, T.: On degenerate q -Bernoulli polynomials. Bull. Korean Math. Soc. 53(4), 1149–1156 (2016)

22. Kim, T.: λ-Analogue of Stirling numbers of the first kind. Adv. Stud. Contemp. Math. (Kyungshang) 27(3), 423–429 (2017)

23. Kim, T.: A note on degenerate Stirling polynomials of the second kind. Proc. Jangjeon Math. Soc. 20(3), 319–331 (2017)

24. Kim, T.: Degenerate Cauchy numbers and polynomials of the second kind. Adv. Stud. Contemp. Math. (Kyungshang) 27(4), 441–449 (2018)

25. Kim, T., Jang, G.-W.: Higher-order degenerate q-Bernoulli polynomials. Proc. Jangjeon Math. Soc. 20(1), 51–60 (2017)

26. Kim, T., Jang, G.W.: A note on degenerate gamma function and degenerate Stirling number of the second kind. Adv. Stud. Contemp. Math. (Kyungshang) 28(2), 207–214 (2018)

27. Kim, T., Kim, D.S.: Degenerate Laplace transform and degenerate gamma function. Russ. J. Math. Phys. 24(2), 241–248 (2017)

28. Kim, T., Kim, D.S.: Identities for degenerate Bernoulli polynomials and Korobov polynomials. Sci. China Math. (2018). http://engine.scichina.com/publisher/scp/journal/SCM/doi/10.1007/s11425-018-9338-5?slug=abstract. https://doi.org/10.1007/s11425-018-9338-5

29. Kim, T., Simsek, Y.: Analytic continuation of the multiple Daehee $$q-l$$-functions associated with Daehee numbers. Russ. J. Math. Phys. 15(1), 58–65 (2008)

30. Kim, T., Yao, Y., Kim, D.S., Jang, G.-W.: Degenerate r-Stirling numbers and r-Bell polynomials. Russ. J. Math. Phys. 25(1), 44–58 (2018)

31. Lim, D.: Modified q-Daehee numbers and polynomials. J. Comput. Anal. Appl. 21(2), 324–330 (2016)

32. Liu, C., Wuyungaowa: Application of probabilistic method on Daehee sequences. Eur. J. Pure Appl. Math. 11(1), 69–78 (2018)

33. Moon, E.-J., Park, J.-W., Rim, S.-H.: A note on the generalized q-Daehee numbers of higher order. Proc. Jangjeon Math. Soc. 17(4), 557–565 (2014)

34. Ozden, H., Cangul, I.N., Simsek, Y.: Remarks on q-Bernoulli numbers associated with Daehee numbers. Adv. Stud. Contemp. Math. (Kyungshang) 18(1), 41–48 (2009)

35. Park, J.-W.: On the q-analogue of Daehee numbers and polynomials. Proc. Jangjeon Math. Soc. 19(3), 537–544 (2016)

36. Park, J.-W., Kim, B.M., Kwon, J.: On a modified degenerate Daehee polynomials and numbers. J. Nonlinear Sci. Appl. 10, 1108–1115 (2017)

37. Pyo, S.-S.: Degenerate Cauchy numbers and polynomials of the fourth kind. Adv. Stud. Contemp. Math. (Kyungshang) 28(1), 127–138 (2018)

38. Rim, S.-H., Kim, T., Pyo, S.-S.: Identities between harmonic, hyperharmonic and Daehee numbers. J. Inequal. Appl. 2018, 168 (2018)

39. Schikhof, W.H.: Ultrametric Calculus: An Introduction to a p-Adic Analysis. Cambridge Studies in Advanced Mathematics, vol. 4, p. 167, Definition 55.1. Cambridge University Press, Cambridge (1985)

40. Shiratani, K., Yokoyama, S.: An application of p-adic convolutions. Mem. Fac. Sci., Kyushu Univ., Ser. A, Math. 36(1), 73–83 (1982)

41. Simsek, Y.: Analysis of the p-adic q-Volkenborn integrals; an approach to generalized Apostrol-type special numbers and polynomials and their applications. Cogent Math. 3, 1269393 (2016)

42. Simsek, Y.: Apostol type Daehee numbers and polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 26(3), 555–566 (2016)

43. Simsek, Y.: Identities on the Changhee numbers and Apostol-type Daehee polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 27(2), 199–212 (2017)

44. Simsek, Y.: Identities and relations related to combinatorial numbers and polynomials. Proc. Jangjeon Math. Soc. 20(1), 127–135 (2017)

45. Simsek, Y.: Construction of some new families of Apostol-type numbers and polynomials via Dirichlet character and p-adic q-integrals. Turk. J. Math. 42, 557–577 (2018)

46. Simsek, Y., Rim, S.-H., Jang, L.-C., Kang, D.-J., Seo, J.-J.: A note on q-Daehee sums. In: Proceedings of the 16th. International Conference of the Jangjeon Mathematical Society, vol. 36, pp. 159–166. Jangjeon Math. Soc., Hapcheon (2005)

47. Simsek, Y., Yardimci, A.: Applications on the Apostol–Daehee numbers and polynomials associated with special numbers, polynomials, and p-adic integrals. Adv. Differ. Equ. 2016, 308 (2016)

48. Washington, L.C.: Introduction to Cyclotomic Fields, 2nd edn. Graduate Texts in Mathematics, vol. 83, xiv+487 pp. Springer, New York (1997). ISBN 0-387947620

## Funding

This paper was supported by Wonkwang University in 2017.

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Correspondence to Lee-Chae Jang.

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