1 Introduction
For a fixed odd prime number p, we make use of the following notation. \(\mathbb{Z}_{p}\), \(\mathbb{Q}_{p}\), and \(\mathbb{C}_{p}\) will denote the ring of p-adic rational integers, the field of p-adic rational numbers and the completions of algebraic closure of \(\mathbb {Q}_{p}\), respectively. The p-adic norm is defined \(|p|_{p} = p^{-1}\) (see [14, 15, 17, 19, 30]).
When one says 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 \(q^{x} = \exp(x\log q)\), \(|x|_{p}\leq1\).
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The q-analog of number x is defined as
Note that \(\lim_{q \rightarrow1} [x]_{q} = x \) for each \(x \in\mathbb{Z}_{p}\).
$$ [x]_{q} =\frac{1- q^{x}}{1-q}. $$
Let \(UD(\mathbb{Z}_{p})= \{f|f:{\mathbb{Z}}_{p} \rightarrow{\mathbb {R}}\text{ is uniformly differentiable} \}\). For \(f\in UD({\mathbb{Z}_{p}})\), the fermionic
p-adic
q-integral on
\({\mathbb{Z}_{p}}\) is defined by Kim as follows (see [9, 14, 15, 17, 19, 20]):
If we put \(f_{1}\) to the translation of f with \(f_{1} (x )=f (x+1 )\), then, by (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 to be
By (3), we have
$$ I_{-q}(f)= \int_{\mathbb{Z}_{p}}f(x)\,d\mu_{-q}(x)=\lim_{N \rightarrow\infty} \frac{1}{[p^{N}]_{-q}}\sum_{x=0}^{p^{N}-1} f(x) (-q)^{x}. $$
(1)
$$ qI_{-q}(f_{1})+I_{-q} (f)=[2]_{q}f(0)\quad \text{(see [1, 17--20, 23, 24, 28, 31])}. $$
(2)
$$ (x )_{n}=x (x-1 )\cdots (x-n+1 )=\sum _{l=0}^{n}S_{1} (n,l )x^{l}, $$
(3)
$$ \bigl(e^{t}-1 \bigr)^{m}=m!\sum _{l=m}^{\infty}S_{2} (l,m )\frac{t^{l}}{l!} \quad \text{(see [3--5, 11])}. $$
(4)
$$ \bigl(\log(1+x) \bigr)^{n}=n!\sum _{l=n} ^{\infty}S_{1}(l,n)\frac {x^{l}}{l!} \quad (n\geq0). $$
(5)
The unsigned Stirling numbers of the first kind are given by
Note that if we replace x to −x in (3), then
(see [3, 5, 28, 31]). Hence \(S_{1}(n,l)=|S_{1}(n,l)|(-1)^{n-l}\).
$$ x^{(n)}=x(x+1)\cdots(x+n-1)=\sum _{l=0} ^{n} \bigl\vert S_{1}(n,l) \bigr\vert x^{l}\quad \text{(see [3, 5])}. $$
(6)
$$\begin{aligned} (-x)_{n}&=(-1)^{n}x^{(n)}=\sum _{l=0} ^{n} S_{1}(n,l) (-1)^{l}x^{l} \\ &=(-1)^{n}\sum_{l=0} ^{n} \bigl\vert S_{1}(n,l) \bigr\vert x^{l} \end{aligned}$$
(7)
In [16], Kim firstly constructed the new \((h,q)\)-extension of the Bernoulli numbers and polynomials with the aid of q-Volkenborn integration, and Simsek gave the Witt-formula for \((h,q)\)-Bernoulli numbers in [27, 34]. Ozden and Simsek defined \((h,q)\)-extension of Euler numbers and polynomials withe the aid of fermionic integral of the function \(f(x)=q^{hx}e^{xt}\) in [29], and found recurrence identities for \((h,q)\)-Euler polynomials and the alternating sums of powers of consecutive \((h,q)\)-integers in [35]. In Chapter 6 of [27], the author discusses several generalizations of Bernoulli numbers and associated polynomials with interpolation at negative integers.
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Kim et al. introduced the Changhee polynomials of the first kind of order
r, defined by the generating function to be
and Moon et al. defined the q-Changhee polynomials of order
r as follows:
By (2), we note that
and thus we see that \(\sum_{n=0} ^{\infty}\operatorname{Ch}_{n} ^{(r)}(x) \frac {t^{n}}{n!}={\int_{\mathbb{Z}_{p}}}(1+t)^{x+y}\,d\mu_{-q}(y)\).
$$ \biggl(\frac{2}{2+t} \biggr)^{r}(1+t)^{x}= \sum_{n=0} ^{\infty}\operatorname{Ch}_{n} ^{(r)}(x) \frac{t^{n}}{n!} \quad \text{(see [12, 13])}, $$
(8)
$$ \biggl(\frac{1+q}{q(1+t)+1} \biggr)^{r}(1+t)^{x}=\sum _{n=0} ^{\infty} \operatorname{Ch}_{n,q} ^{(r)}(x)\frac{t^{n}}{n!}\quad \text{(see [28, 31])}. $$
$$ \biggl(\frac{1+q}{q(1+t)+1} \biggr)^{r}(1+t)^{x}={ \int_{\mathbb {Z}_{p}}}(1+t)^{x+y}\,d\mu_{-q}(y), $$
In [31], the authors defined the generalization of the q-Changhee polynomials which are called by \((h,q)\)-Changhee polynomials of the first kind and \((h,q)\)-Changhee polynomials of the second kind, respectively, defined by the fermionic p-adic q-integral on \({\mathbb{Z}}_{p}\) to be
$$\begin{aligned}& \begin{aligned}[b] \sum_{n=0} ^{\infty}\operatorname{Ch}_{n,h,q} (x)\frac{t^{n}}{n!}&={ \int_{\mathbb {Z}_{p}}}q^{h}y(1+t)^{x+y}\,d \mu_{-q}(y) \\ &=\frac{1+q}{q^{h+1}(1+t)+1}(1+t)^{x}, \end{aligned} \end{aligned}$$
(9)
$$\begin{aligned}& \begin{aligned}[b] \sum_{n=0} ^{\infty}{\widehat{\operatorname{Ch}}}_{n,h,q} (x) \frac{t^{n}}{n!}&={ \int _{\mathbb{Z}_{p}}}q^{hy}(1+t)^{-x-y}\,d \mu_{-q}(y) \\ &=\frac{1+q}{q^{h+1}+1+t}(1+t)^{r-x}. \end{aligned} \end{aligned}$$
(10)
As is well known, the Euler polynomials are defined by the generating function to be
In [4], Carlitz first introduced the concept of degenerate numbers and polynomials which are related to Euler polynomials as follows:
where \(\lambda\in{\mathbb{R}}\). Note that, by (11), we see that
and thus we get
$$ \frac{2}{e^{t}+1}e^{xt}=\sum_{n=0} ^{\infty} E_{n}(x)\frac {t^{n}}{n!}\quad \text{(see [2, 7, 8, 17, 19, 20, 33, 36, 37])}. $$
$$ \sum_{n=0} ^{\infty}E_{n} (x|\lambda) \frac{t^{n}}{n!}=\frac {2}{(1+\lambda t)^{\frac{1}{\lambda}}+1}(1+\lambda t)^{\frac {x}{\lambda}}, $$
(11)
$$\begin{aligned} \lim_{\lambda\rightarrow0}\sum_{n=0} ^{\infty}E_{n} (x|\lambda )\frac{t^{n}}{n!}&=\lim _{\lambda\rightarrow0}\frac{2}{(1+\lambda t)^{\frac{1}{\lambda}}+1}(1+\lambda t)^{\frac{x}{\lambda}} \\ &=\frac{2}{e^{t}+1}e^{xt}=\sum_{n=0} ^{\infty}E_{n} (x)\frac{t^{n}}{n!}, \end{aligned}$$
$$ \lim_{\lambda\rightarrow0}E_{n} (x|\lambda)=E_{n}(x). $$
In the recent years, the degenerate of some special polynomials are investigated by many authors (see [4, 10, 11, 21‐24, 26]). In particular, the degenerate Changhee polynomials which are defined by the generating function to be
and Kim et al. defined the degenerate
q-Changhee polynomials as follows:
$$ \sum_{n=0} ^{\infty} \operatorname{Ch}_{n,\lambda} ^{*} (x)\frac{t^{n}}{n!}= \frac {2\lambda}{2\lambda+\log(1+\lambda t)} \bigl(1+\log(1+\lambda )^{\frac{1}{\lambda t}} \bigr)^{x} \quad \text{(see [26])}, $$
(12)
$$ \sum_{n=0} ^{\infty} \operatorname{Ch}_{n,\lambda,q}(x) \frac{t^{n}}{n!}=\frac {q\lambda+\lambda}{q\log(1+\lambda t)+q\lambda+\lambda} \bigl(1+\log(1+\lambda t)^{\frac{1}{\lambda}} \bigr)^{x}\quad \text{(see [22, 24])}. $$
(13)
In the past decade, many researchers have investigated the various generalization of Changhee polynomials (see [1, 6, 12, 13, 24‐26, 28, 31]), and in [1, 31], the authors gave new q-analog of Changhee numbers and polynomials.
In this paper, we introduce a new q-analog of degenerate Changhee numbers and polynomials of the first kind and the second kind of order r, and derive some new interesting identities related to the degenerate q-Changhee polynomials of order r.
2 q-Analog of degenerate Changhee polynomials
Let assume that \(\lambda,t \in{\mathbb{C}_{p}}\) with \(|\lambda t|< p^{-\frac{1}{p-1}}\). By (2), we get
where \(h\in{\mathbb{Z}}\). By (14), we define the q-analog of degenerate Changhee polynomials by the generating function to be
In the special case \(x=0\), \(\operatorname{Ch}_{n,h,q} (\lambda)=\operatorname{Ch}_{n,h,q} (0|\lambda )\) are called the q-analog of degenerate Changhee numbers.
$$\begin{aligned}& { \int_{\mathbb{Z}_{p}}}q^{hy} \biggl(1+\frac{1}{\lambda}\log (1+ \lambda t) \biggr)^{x+y}\,d\mu_{-q}(y) \\& \quad =\frac{1+q}{q^{h+1} (1+\frac{1}{\lambda}\log(1+\lambda t) )+1} \biggl(1+\frac {1}{\lambda}\log(1+\lambda t) \biggr)^{x}, \end{aligned}$$
(14)
$$ \sum_{n=0} ^{\infty} \operatorname{Ch} _{n,h,q}(x|\lambda)\frac{t^{n}}{n!}=\frac {1+q}{q^{h+1} (1+\frac{1}{\lambda}\log(1+\lambda t) )+1} \biggl(1+\frac{1}{\lambda}\log(1+\lambda t) \biggr)^{x}. $$
(15)
Note that
and so we see that
and, if we put \(h=0\), then
$$\begin{aligned} \lim_{\lambda\rightarrow0} \sum_{n=0} ^{\infty}\operatorname{Ch} _{n,h,q}(x|\lambda)\frac{t^{n}}{n!} &= \lim_{\lambda\rightarrow0} \frac{1+q}{q^{h+1} (1+\frac {1}{\lambda}\log(1+\lambda t) )+1} \biggl(1+\frac{1}{\lambda } \log(1+\lambda t) \biggr)^{x} \\ &=\frac{q+1}{q^{h+1}(1+t)+1}(1+t)^{x}=\sum_{n=0} ^{\infty} \operatorname{Ch}_{n,h,q} (x)\frac{t^{n}}{n!}, \end{aligned}$$
$$ \lim_{\lambda\rightarrow0} \operatorname{Ch}_{n,h,q} (x|\lambda)=\operatorname{Ch}_{n,h,q} (x), $$
(16)
$$ \lim_{\lambda\rightarrow0}\operatorname{Ch}_{n,q} ^{(0)}(x|\lambda )=\operatorname{Ch}_{n,q}(x)\quad \text{and} \quad \operatorname{Ch}_{n,0,q}(x|\lambda)=\operatorname{Ch}_{n,\lambda,q}(x). $$
(17)
By (16) and (17), we see that q-analog of degenerate Changhee polynomials are closely related to the q-Changhee polynomials and degenerate q-Changhee polynomials.
By using (7) and (14), we have
By (14) and (18), we have
By (3), we get
where \(E_{n} (x|h,q)\) is the nth \((h,q)\)-Euler polynomials which are defined by the generating function to be
$$\begin{aligned}& { \int_{\mathbb{Z}_{p}}}q^{hy} \biggl(1+\frac{1}{\lambda}\log (1+ \lambda t) \biggr)^{x+y}\,d\mu_{-q}(y) \\& \quad = { \int_{\mathbb{Z}_{p}}}q^{hy} \sum_{n=0} ^{\infty}\binom {x+y}{n}\lambda^{-n} \bigl(\log(1+\lambda t) \bigr)^{n}\,d\mu_{-q}(y) \\& \quad = { \int_{\mathbb{Z}_{p}}}q^{hy}\sum_{n=0} ^{\infty}\binom {x+y}{n}\lambda^{-n}n!\sum _{l=n} ^{\infty}S_{1}(l,n)\frac{(\lambda t)^{l}}{l!} \,d\mu_{-q}(y) \\& \quad = \sum_{n=0} ^{\infty}\sum _{m=0} ^{n} \lambda^{n-m}m!S_{1}(n,m){ \int _{\mathbb{Z}_{p}}}q^{hy}\binom{x+y}{m}\,d \mu_{-q}(y)\frac{t^{n}}{n!} \\& \quad = \sum_{n=0} ^{\infty} \Biggl(\sum _{m=0} ^{n} \lambda ^{n-m}S_{1}(n,m){ \int_{\mathbb{Z}_{p}}}q^{hy} (x+y)_{m}\,d\mu _{-q}(y) \Biggr)\frac{t^{n}}{n!}. \end{aligned}$$
(18)
$$ \operatorname{Ch}_{n,h,q} (x|\lambda)=\sum _{m=0} ^{n} \lambda^{n-m}S_{1}(n,m){ \int _{\mathbb{Z}_{p}}}q^{hy} (x+y)_{m}\,d \mu_{-q}(y). $$
(19)
$$\begin{aligned} { \int_{\mathbb{Z}_{p}}}q^{hy} (x+y)_{m}\,d \mu_{-q}(y) &=\sum_{l=0} ^{m} S_{1}(m,l) { \int_{\mathbb{Z}_{p}}}q^{hy} (x+y)^{l} \,d\mu _{-q}(y) \\ &=\sum_{l=0} ^{m} S_{1}(m,l)E_{l} (x|h,q), \end{aligned}$$
(20)
$$\begin{aligned} \sum_{n=0} ^{\infty} E_{n} (x|h,q) \frac{t^{n}}{n!}&={ \int_{\mathbb {Z}_{p}}}q^{hy}e^{t(x+y)}\,d \mu_{-q}(y) \\ &=\sum_{n=0} ^{\infty} \biggl({ \int_{\mathbb {Z}_{p}}}q^{hy}(x+y)^{n}\,d \mu_{-q}(y) \biggr)\frac{t^{n}}{n!} \\ &=\frac{q+1}{q^{h+1}e^{t}+1}e^{xt}\quad \text{(see [32])}. \end{aligned}$$
In addition,
By (19), (20) and (21), we obtain the following theorem.
$$\begin{aligned}& \frac{q+1}{q^{h+1} (1+\frac{1}{\lambda}\log(1+\lambda t) )+1} \\& \quad = (q+1)\sum_{m=0} ^{\infty}(-1)^{m}q^{m(h+1)} \biggl(1+\frac{1}{\lambda }\log(1+\lambda t) \biggr)^{m} \\& \quad = (q+1)\sum_{m=0} ^{\infty}(1-)^{m}q^{m(h+1)} \sum_{l=0} ^{m} \lambda ^{-l} \binom{m}{l} \bigl(\log(1+\lambda t) \bigr)^{l} \\& \quad = (q+1)\sum_{m=0} ^{\infty}q^{m(h+1)}(-1)^{m} \sum_{l=0} ^{m}\lambda ^{-l} \binom{m}{l}l!\sum_{r=l} ^{\infty}S_{1}(r,l) \lambda^{r}\frac {t^{r}}{r!} \\& \quad = (q+1)\sum_{n=0} ^{\infty}\sum _{l=0} ^{\infty }q^{(n+l)(h+1)}(-1)^{n+l} \binom{n+l}{l}\sum_{r=0} ^{\infty}\lambda ^{r} S_{1}(r+l,l)\frac{t^{r+l}}{(r+l)!} \\& \quad = \sum_{n=0} ^{\infty} \Biggl(\sum _{m=0} ^{\infty}\sum _{l=0} ^{n}q^{(m+l)(h+1)}(-1)^{m+l} \binom{m+l}{l}\lambda^{n-l}S_{1}(n,l) \Biggr) \frac{t^{n}}{n!}. \end{aligned}$$
(21)
Theorem 2.1
For each nonnegative integer
n, we have
and
$$\begin{aligned} \operatorname{Ch}_{n,h,q} (x|\lambda)&= \sum _{m=0} ^{n} \lambda^{n-m}S_{1}(n,m){ \int _{\mathbb{Z}_{p}}}q^{hy} (x+y)_{m}\,d \mu_{-q}(y) \\ &=\sum_{m=0} ^{n} \sum _{l=0} ^{m}\lambda^{n-m}S_{1}(n,m)S_{1}(m,l)E_{l} (x|h,q) \end{aligned}$$
$$ \operatorname{Ch}_{n,h,q} (\lambda)=\sum_{m=0} ^{\infty}\sum_{l=0} ^{n}q^{(m+l)(h+1)}(-1)^{m+l} \binom{m+l}{l}\lambda^{n-l}S_{1}(n,l). $$
By replacing t by \(\frac{1}{\lambda} (e^{\lambda t}-1 )\) in (15) and by using (4), we have
and, thus, by (9) and (22), we have the following corollary.
$$\begin{aligned} \frac{q+1}{q^{h+1}(1+t)+1}(1+t)^{x}&=\sum_{n=0} ^{\infty} \operatorname{Ch}_{n,q} ^{(h)} (x|\lambda) \frac{1}{n!} \biggl(\frac{1}{\lambda} \bigl(e^{\lambda t}-1 \bigr) \biggr)^{n} \\ &=\sum_{n=0} ^{\infty}\operatorname{Ch}_{n,q} ^{(h)}(x|\lambda)\lambda^{-n}\frac {(e^{t}-1)^{n}}{\lambda} \\ &= \Biggl(\sum_{n=0} ^{\infty} \operatorname{Ch}_{n,q} ^{(h)}(x|\lambda)\lambda ^{-n}\frac{1}{n!} \Biggr) \Biggl(n!\sum _{l=n} ^{\infty}S_{2}(l,n)\frac {(\lambda t)^{l}}{l!} \Biggr) \\ &=\sum_{n=0} ^{\infty} \Biggl(\sum _{m=0} ^{n}\lambda^{n-m} \operatorname{Ch}_{m,q} ^{(h)}(x|\lambda)S_{2}(n,m) \Biggr)\frac{t^{n}}{n!}, \end{aligned}$$
(22)
Corollary 2.2
For each nonnegative integer
n, we have
$$ \operatorname{Ch}_{n,h,q} (x)=\sum_{m=0} ^{n}\lambda^{n-m}S_{2}(n,m) \operatorname{Ch}_{m,h,q} (x|\lambda). $$
From (1) and (14), we note that
By (23), we obtain the following theorem.
$$\begin{aligned}& \sum_{n=0} ^{\infty} \bigl(q^{h+1} \operatorname{Ch}_{n,h,q} (x+1|\lambda )+\operatorname{Ch}_{n,h,q} (x|\lambda) \bigr)\frac{t^{n}}{n!} \\& \quad = \frac{(q+1)}{q^{h+1} (1+\frac{1}{\lambda}\log(1+\lambda t) )+1} \biggl(q^{h+1} \biggl(1+\frac{1}{\lambda} \log(1+\lambda t) \biggr)+1 \biggr) \biggl(1+\frac{1}{\lambda}\log(1+\lambda t) \biggr)^{x} \\& \quad = (q+1) \biggl(1+\frac{1}{\lambda}\log(1+\lambda t) \biggr)^{x} \\& \quad = (q+1)\sum_{n=0} ^{\infty} \Biggl(\sum _{m=0} ^{n} (x)_{m}\lambda ^{n-m}S_{1}(n,m) \Biggr)\frac{t^{n}}{n!}. \end{aligned}$$
(23)
Theorem 2.3
For each nonnegative integer
n, we get
$$ q^{h+1}\operatorname{Ch}_{n,h,q} (x+1|\lambda)+ \operatorname{Ch}_{n,h,q} (x|\lambda)=(q+1)\sum _{m=0} ^{n} (x)_{m}\lambda^{n-m}S_{1}(n,m). $$
For positive integer d with \(d\equiv1\ (\operatorname{mod} 2)\), if we put \(f(x)=q^{hx} (1+\frac{1}{\lambda}\log(1+\lambda t) )^{x}\), then, by (2) and (7), we have
and
$$\begin{aligned}& q^{d} { \int_{\mathbb{Z}_{p}}}q^{h(x+d)} \biggl(1+\frac{1}{\lambda }\log(1+ \lambda t) \biggr)^{x+d}\,d\mu_{-q}(x)+{ \int_{\mathbb {Z}_{p}}}q^{hx} \biggl(1+\frac{1}{\lambda}\log(1+ \lambda t) \biggr)^{x}\,d\mu_{-q}(x) \\& \quad = (q+1)\sum_{l=0} ^{d-1} (-1)^{l}q^{l(h+1)} \biggl(1+\frac{1}{\lambda }\log(1+\lambda t) \biggr)^{l} \\& \quad = (q+1)\sum_{l=0} ^{d-1}(-1)^{l}q^{l(h+1)} \sum_{k=0} ^{\infty}\binom {l}{k} \biggl( \frac{1}{\lambda}\log(1+\lambda t) \biggr)^{k} \\& \quad = (q+1)\sum_{l=0} ^{d-1}(-1)^{l}q^{l(h+1)} \sum_{k=0} ^{\infty}\binom {l}{k} \lambda^{-k}\sum_{r=k} ^{\infty}S_{1}(r,k) \frac{t^{r}}{r!} \\& \quad = \sum_{n=0} ^{\infty} \Biggl((q+1)\sum _{l=0} ^{d-1}\sum _{k=0} ^{n}(-1)^{l}q^{l(h+1)} \binom{l}{k}\lambda^{-k}S_{1}(n,k) \Biggr) \frac{t^{n}}{n!} \end{aligned}$$
(24)
$$\begin{aligned}& q^{d} { \int_{\mathbb{Z}_{p}}}q^{h(x+d)} \biggl(1+\frac{1}{\lambda }\log(1+ \lambda t) \biggr)^{x+d}\,d\mu_{-q}(x) \\& \qquad {}+{ \int_{\mathbb {Z}_{p}}}q^{hx} \biggl(1+\frac{1}{\lambda}\log(1+ \lambda t) \biggr)^{x}\,d\mu_{-q}(x) \\& \quad =\sum_{n=0} ^{\infty} \bigl(q^{d(h+1)} \operatorname{Ch}_{n,q} ^{(h)}(d|\lambda )+\operatorname{Ch}_{n,q} ^{(h)}(\lambda) \bigr) \frac{t^{n}}{n!}. \end{aligned}$$
(25)
Theorem 2.4
For each nonnegative integer
n
and odd integer
d, we have
$$ q^{d(h+1)} \operatorname{Ch}_{n,h,q} (d|\lambda)+ \operatorname{Ch}_{n,h,q} (\lambda)=(q+1)\sum _{l=0} ^{d-1}\sum_{k=0} ^{n}(-1)^{l}q^{l(h+1)}\binom{l}{k} \lambda^{-k}S_{1}(n,k). $$
3 q-Analog of higher order degenerate Changhee polynomials
In this section, we consider the q-analog of higher order degenerate Changhee polynomials which are defined by
where n is a nonnegative integer, \(h_{1},\ldots,h_{r}\in{\mathbb{Z}}\) and \(r\in{\mathbb{N}}\).
$$\begin{aligned}& \operatorname{Ch}_{n,q} ^{(h_{1},\ldots,h_{r})}(x|\lambda) \\& \quad =\sum_{m=0} ^{n} \lambda^{n-m}S_{1}(n,m)\underbrace{{ \int_{\mathbb {Z}_{p}}}\cdots{ \int_{\mathbb{Z}_{p}}}}_{r\text{-times}} q^{\sum_{i=1} ^{r}h_{i}y_{i}}(x+y_{1} \\& \qquad {}+ \cdots+y_{r})_{m}\,d\mu_{-q}(y_{1})\cdots \,d\mu_{-q}(y_{r}), \end{aligned}$$
(26)
By (26), we have
By (26) and (27), we see that
If we put
then
and
Thus, \(F_{q} ^{(h_{1},\ldots,h_{r})}(x,t)\) seems to be a new q-extension of the generating function for the degenerate Changhee polynomials of order r.
$$\begin{aligned}& \sum_{n=0} ^{\infty} \Biggl(\sum _{m=0} ^{n}\lambda ^{n-m}S_{1}(n,m){ \int_{\mathbb{Z}_{p}}}\cdots{ \int_{\mathbb {Z}_{p}}}q^{\sum_{i=1} ^{r}h_{i}y_{i}}(x+y_{1}+ \cdots+y_{r})_{m}\,d\mu _{-q}(y_{1}) \cdots \,d\mu_{-q}(y_{r}) \Biggr)\frac{t^{n}}{n!} \\& \quad = \sum_{n=0} ^{\infty} \sum _{m=0} ^{n} \lambda^{n-m}m!S_{1}(n,m){ \int _{\mathbb{Z}_{p}}}\cdots{ \int_{\mathbb{Z}_{p}}}q^{\sum_{i=1} ^{r} h_{i}y_{i}} \\& \qquad {}\times\binom{x+y_{1}+\cdots+y_{r}}{m}\,d \mu_{-q}(y_{1})\cdots \,d\mu _{-q}(y_{r}) \frac{t^{n}}{n!} \\& \quad = { \int_{\mathbb{Z}_{p}}}\cdots{ \int_{\mathbb{Z}_{p}}}q^{\sum _{i=1} ^{r} h_{i}y_{i}}\sum_{n=0} ^{\infty}\binom{x+y_{1}+\cdots +y_{r}}{n}\lambda^{-n}n! \\& \qquad {}\times\sum _{l=n} ^{\infty}S_{1}(l,n)\frac{(\lambda t)^{l}}{l!} \,d\mu_{-q}(y_{1})\cdots \,d\mu_{-q}(y_{r}) \\& \quad = { \int_{\mathbb{Z}_{p}}}\cdots{ \int_{\mathbb{Z}_{p}}}q^{\sum _{i=1} ^{r} h_{i}y_{i}}\sum_{n=0} ^{\infty}\binom{x+y_{1}+\cdots +y_{r}}{n}\lambda^{-n} \bigl(\log(1+\lambda t) \bigr)^{n} \,d\mu _{-q}(y_{1})\cdots \,d \mu_{-q}(y_{r}) \\& \quad = { \int_{\mathbb{Z}_{p}}}\cdots{ \int_{\mathbb{Z}_{p}}}q^{\sum _{i=1} ^{r} h_{i}y_{i}} \biggl(1+\frac{1}{\lambda}\log(1+ \lambda t) \biggr)^{x+y_{1}+\cdots+y_{r}}\,d\mu_{-q}(y_{1})\cdots \,d \mu_{-q}(y_{r}) \\& \quad = \prod_{i=1} ^{r} \biggl( \frac{1+q}{q^{h_{i}+1} (1+\frac{1}{\lambda }\log(1+\lambda t) )+1} \biggr) \biggl(1+\frac{1}{\lambda}\log (1+\lambda t) \biggr)^{x}. \end{aligned}$$
(27)
$$ \sum_{n=0} ^{\infty} \operatorname{Ch}_{n,q} ^{(h_{1},\ldots,h_{r})}(x|\lambda)\frac{t^{n}}{n!}= \prod_{i=1} ^{r} \biggl(\frac{1+q}{q^{h_{i}+1} (1+\frac{1}{\lambda }\log(1+\lambda t) )+1} \biggr) \biggl(1+\frac{1}{\lambda}\log (1+\lambda t) \biggr)^{x}. $$
(28)
$$ F_{q} ^{(h_{1},\ldots,h_{r})}(x,t)=\prod_{i=1} ^{r} \biggl(\frac {1+q}{q^{h_{i}+1} (1+\frac{1}{\lambda}\log(1+\lambda t) )+1} \biggr) \biggl(1+\frac{1}{\lambda} \log(1+\lambda t) \biggr)^{x}, $$
$$\begin{aligned} F_{q} ^{(0,\ldots,0)}(x,t)&= \biggl(\frac{[2]_{q}}{[2]_{q}+\frac{q}{\lambda }\log(1+\lambda t)} \biggr)^{r} \biggl(1+\frac{1}{\lambda}\log (1+\lambda t) \biggr)^{x} \\ &=\sum_{n=0} ^{\infty}\operatorname{Ch}_{n,\lambda,q} ^{(r)}(x)\frac{t^{n}}{n!} \end{aligned}$$
$$\begin{aligned} \lim_{q\rightarrow1}F_{q} ^{(-1,\ldots,-1)}(x,t)&= \biggl( \frac {2}{\frac{1}{\lambda}\log(1+\lambda t)+2} \biggr)^{r} \biggl(1+\frac {1}{\lambda}\log(1+\lambda t) \biggr)^{x} \\ &=\sum_{n=0} ^{\infty} \operatorname{Ch}_{n,\lambda} ^{(r)} (x)\frac{t^{n}}{n!}. \end{aligned}$$
Note that
$$\begin{aligned}& \prod_{i=1} ^{r} \biggl(\frac{1+q}{q^{h_{i}+1} (1+\frac{1}{\lambda }\log(1+\lambda t) )+1} \biggr) \biggl(1+\frac{1}{\lambda}\log (1+\lambda t) \biggr)^{x} \\& \quad = \Biggl(\sum_{n=0} ^{\infty} \operatorname{Ch}_{n,q} ^{(h_{1},\ldots,h_{r})}(\lambda )\frac{t^{n}}{n!} \Biggr) \Biggl(\sum_{n=0} ^{\infty}\sum _{m=0} ^{n} \binom{x}{m}\lambda^{n-m}m!S_{1}(n,m) \frac{t^{n}}{n!} \Biggr) \\& \quad = \sum_{n=0} ^{\infty} \Biggl(\sum _{m=0} ^{n}\sum _{l=0} ^{n-m} \binom {x}{l}\binom{n}{m} \lambda^{n-m-l}l!\operatorname{Ch}_{m,q} ^{(h_{1},\ldots ,h_{r})}( \lambda)S_{1}(n-m,l) \Biggr)\frac{t^{n}}{n!}. \end{aligned}$$
(29)
Since
where \(\binom{n}{l_{1},l_{2},\ldots,l_{r}}=\frac{n!}{l_{1}!l_{2}!\cdots l_{r}!}\), we have
where \(E_{n} (h,q)=E_{n} (0|h,q)\), which are called the \((h,q)\)-Euler numbers.
$$\begin{aligned}& (x+y_{1}+\cdots+y_{r})_{n} \\& \quad =\sum_{l=0} ^{n} S_{1}(n,l) (x+y_{1}+\cdots+y_{r})^{l} \\& \quad =\sum_{l=0} ^{n} S_{1}(n,l) \sum_{\substack{l_{1}+\cdots+l_{r}=l\\ l_{1},\ldots,l_{r}\geq0}} \binom{n}{l_{1},l_{2},\ldots,l_{r}}y_{1} ^{l_{1}}y_{2} ^{l_{2}} \cdots(x+y_{r}) ^{l_{r}}, \end{aligned}$$
(30)
$$\begin{aligned}& { \int_{\mathbb{Z}_{p}}}\cdots{ \int_{\mathbb{Z}_{p}}}q^{\sum _{i=1} ^{r}h_{i}y_{i}}(x+y_{1}+ \cdots+y_{r})_{m}\,d\mu_{-q}(y_{1})\cdots \,d\mu _{-q}(y_{r}) \\& \quad = { \int_{\mathbb{Z}_{p}}}\cdots{ \int_{\mathbb{Z}_{p}}}q^{\sum _{i=1} ^{r}h_{i}y_{i}}\sum_{l=0} ^{m} S_{1}(m,l) (x+y_{1}+\cdots+y_{r})^{l} \,d\mu _{-q}(y_{1})\cdots \,d\mu_{-q}(y_{r}) \\& \quad = { \int_{\mathbb{Z}_{p}}}\cdots{ \int_{\mathbb{Z}_{p}}}q^{\sum _{i=1} ^{r}h_{i}y_{i}}\sum_{l=0} ^{m} S_{1}(m,l) \\& \qquad {}\times\sum_{\substack{l_{1}+\cdots +l_{r}=l\\l_{1},\ldots,l_{r}\geq0}} \binom{m}{l_{1},l_{2},\ldots,l_{r}}y_{1} ^{l_{1}}y_{2} ^{l_{2}} \cdots(x+y_{r}) ^{l_{r}} \,d\mu_{-q}(y_{1}) \cdots \,d\mu _{-q}(y_{r}) \\& \quad = \sum_{l=0} ^{m} S_{1}(m,l)\sum_{\substack{l_{1}+\cdots+l_{r}=l \\l_{1},\ldots,l_{r}\geq0}}\binom{m}{l_{1},l_{2},\ldots,l_{r}}E_{l_{1}} (h_{1},q)\cdots E_{l_{r-1}} (h_{r-1},q)E_{l_{r},q} (x|h_{r},q), \end{aligned}$$
(31)
Theorem 3.1
For
\(n \geq0\), we have
$$\begin{aligned}& \operatorname{Ch}_{n,q} ^{(h_{1},\ldots,h_{r})}(x|\lambda) \\& \quad=\sum_{m=0} ^{n}\sum _{l=0} ^{m}\sum_{\substack{l_{1}+\cdots+l_{r}=l\\ l_{1},\ldots,l_{r}\geq0}} \lambda^{n-m}S_{1}(n,m) S_{1}(m,l) \binom {m}{l_{1},l_{2},\ldots,l_{r}} \\& \qquad {}\times E_{l_{1}} (h_{1},q)\cdots E_{l_{r-1}} (h_{r-1},q)E_{l_{r},q} (x|h_{r},q). \end{aligned}$$
By replacing t by \(\frac{1}{\lambda} (e^{\lambda t}-1 )\) in (28),
and, by (9),
Thus, by (32) and (33), we obtain the following theorem.
$$\begin{aligned}& \sum_{n=0} ^{\infty}\operatorname{Ch}_{n,q} ^{(h_{1},\ldots,h_{r})}(x|\lambda)\frac {1}{n!} \biggl(\frac{1}{\lambda} \bigl(e^{\lambda t}-1 \bigr) \biggr)^{n} \\& \quad = \sum_{n=0} ^{\infty} \operatorname{Ch}_{n,q} ^{(h_{1},\ldots,h_{r})}(x|\lambda )\frac{1}{n!} \lambda^{-n}n!\sum_{l=n} ^{\infty}S_{2}(l,n)\frac {(\lambda t)^{l}}{l!} \\& \quad =\sum_{n=0} ^{\infty} \Biggl(\sum _{m=0} ^{n} \operatorname{Ch}_{m,q} ^{(h_{1},\ldots ,h_{r})}(x|\lambda)\lambda^{n-m}S_{2}(n,m) \Biggr) \frac{t^{n}}{n!} \end{aligned}$$
(32)
$$\begin{aligned}& \prod_{i=1} ^{r} \biggl(\frac{q+1}{q^{h_{i}+1}(1+t)+1} \biggr) (1+t)^{x} \\& \quad = \Biggl(\prod_{i=1} ^{r-1} \Biggl( \sum_{n=0} ^{\infty}\operatorname{Ch}_{n,q} ^{(h_{i})}\frac{t^{n}}{n!} \Biggr) \Biggr) \Biggl(\sum _{n=0} ^{\infty }\operatorname{Ch}_{n,q} ^{(h_{r})}(x)\frac{t^{n}}{n!} \Biggr) \\& \quad = \sum_{n=0} ^{\infty}\sum _{\substack{l_{1}+\cdots+l_{r}=n \\l_{1},\ldots ,l_{r}\geq0}}\binom{n}{l_{1},\ldots,l_{r}}\operatorname{Ch}_{l_{1},q} ^{(h_{1})}\cdots \operatorname{Ch}_{l_{r-1},q} ^{(h_{r-1})} \operatorname{Ch}_{l_{r},q} ^{(h_{r})}(x)\frac{t^{n}}{n!}. \end{aligned}$$
(33)
Theorem 3.2
For
\(n\geq0\), we have
$$ \sum_{\substack{l_{1}+\cdots+l_{r}=n\\l_{1},\ldots,l_{r}\geq0}}\binom {n}{l_{1},\ldots,l_{r}}\operatorname{Ch}_{l_{1},q} ^{(h_{1})}\cdots \operatorname{Ch}_{l_{r-1},q} ^{(h_{r-1})} \operatorname{Ch}_{l_{r},q} ^{(h_{r})}(x)=\sum _{m=0} ^{n} \operatorname{Ch}_{m,q} ^{(h_{1},\ldots,h_{r})}(x|\lambda)\lambda^{n-m}S_{2}(n,m). $$
4 q-Analog of higher order degenerate Changhee polynomials of the second kind
In this section, we consider the q-analog of higher order degenerate Changhee polynomials of the second kind is defined as follows:
where n is a nonnegative integer. In particular, \(\widehat{\operatorname{Ch}}_{n,q} ^{(h_{1},\ldots,h_{r})}(0|\lambda)=\widehat{\operatorname{Ch}}_{n,q} ^{(h_{1},\ldots ,h_{r})}(\lambda)\) are called the q-analog of higher order degenerate Changhee numbers of the second kind.
$$\begin{aligned}& \widehat{\operatorname{Ch}}_{n,q} ^{(h_{1},\ldots,h_{r})}(x|\lambda) \\& \quad =\sum_{m=0} ^{n} \lambda^{n-m}S_{1}(n,m)\underbrace{{ \int_{\mathbb {Z}_{p}}}\cdots{ \int_{\mathbb{Z}_{p}}}}_{r\text{-times}} q^{\sum_{i=1} ^{r}h_{i}y_{i}}(-x-y_{1} \\& \qquad {}- \cdots-y_{r})_{m}\,d\mu_{-q}(y_{1})\cdots \,d\mu_{-q}(y_{r}), \end{aligned}$$
(34)
By (7) and (34), it leads to
Thus, we state the following theorem.
$$\begin{aligned}& \widehat{\operatorname{Ch}}_{n,q} ^{(h_{1},\ldots,h_{r})}(x|\lambda) \\& \quad = \sum_{m=0} ^{n} \lambda^{n-m}S_{1}(n,m){ \int_{\mathbb{Z}_{p}}}\cdots { \int_{\mathbb{Z}_{p}}}q^{\sum_{i=1} ^{r}h_{i}y_{i}}(-x-y_{1}-\cdots -y_{r})_{m}\,d\mu_{-q}(y_{1})\cdots \,d \mu_{-q}(y_{r}) \\& \quad = \sum_{m=0} ^{n} \lambda^{n-m}S_{1}(n,m){ \int_{\mathbb{Z}_{p}}}\cdots { \int_{\mathbb{Z}_{p}}}q^{\sum_{i=1} ^{r}h_{i}y_{i}}(-1)^{m} \\& \qquad {}\times (x+y_{1}+\cdots +y_{r})^{(m)}\,d \mu_{-q}(y_{1})\cdots \,d\mu_{-q}(y_{r}) \\& \quad = \sum_{m=0} ^{n} \lambda^{n-m}S_{1}(n,m){ \int_{\mathbb{Z}_{p}}}\cdots { \int_{\mathbb{Z}_{p}}}q^{\sum_{i=1} ^{r}h_{i}y_{i}}(-1)^{m} \\& \qquad {}\times\sum _{l=0} ^{m}\bigl|S_{1}(m,l)\bigr|(x+y_{1}+ \cdots+y_{r})^{l} \,d\mu_{-q}(y_{1}) \cdots \,d\mu _{-q}(y_{r}) \\& \quad = \sum_{m=0} ^{n} \sum _{l=0} ^{m} \lambda^{n-m}S_{1}(n,m)\bigl|S_{1}(m,l)\bigr|(-1)^{m} \\& \qquad {} \times\sum_{\substack{l_{1}+\cdots+l_{r}=l\\l_{1},\ldots,l_{r}\geq 0}}\binom{l}{l_{1},\ldots,l_{r}}\prod _{i=1} ^{r-1}{ \int_{\mathbb {Z}_{p}}}q^{h_{i}y_{i}}y_{i} ^{l_{i}}\,d \mu_{-q}(y_{i}){ \int_{\mathbb {Z}_{p}}}q^{h_{r}y_{r}}(x+y_{r}) ^{l_{r}} \,d\mu_{-q}(y_{r}) \\& \quad = \sum_{m=0} ^{n} \sum _{l=0} ^{m} \sum_{\substack{l_{1}+\cdots+l_{r}=l \\l_{1},\ldots,l_{r}\geq0}} \binom{l}{l_{1},\ldots,l_{r}}\lambda ^{n-m}(-1)^{l} \\& \qquad {}\times S_{1}(n,m)S_{1}(m,l) \Biggl(\prod_{i=1} ^{r-1} E_{l_{i}} ^{(h_{i})}(q) \Biggr)E_{l_{r}} ^{(h_{r})}(x|q). \end{aligned}$$
(35)
Theorem 4.1
For
\(n \geq0\), we have
$$\begin{aligned} \widehat{\operatorname{Ch}}_{n,q} ^{(h_{1},\ldots,h_{r})}(x|\lambda) =&\sum _{m=0} ^{n} \sum _{l=0} ^{m} \sum_{\substack{l_{1}+\cdots+l_{r}=l\\l_{1},\ldots,l_{r}\geq 0}} \binom{l}{l_{1},\ldots,l_{r}}\lambda^{n-m}(-1)^{l}S_{1}(n,m)S_{1}(m,l) \\ &{}\times \Biggl(\prod_{i=1} ^{r-1} E_{l_{i}} (h_{i},q) \Biggr)E_{l_{r}} (x|h_{r},q). \end{aligned}$$
Now, we consider the generating function of the q-analog of higher order degenerate Changhee polynomials of the second kind as follows:
In the special case \(r=1\),
\({\widehat{\operatorname{Ch}}}_{n,q} ^{(h)}(x|\lambda)={\widehat {\operatorname{Ch}}}_{n,q}(x|h,\lambda)\) are called the q-analog of degenerate Changhee polynomials of the second kind.
$$\begin{aligned}& \sum_{n=0} ^{\infty}\widehat{ \operatorname{Ch}}_{n,q} ^{(h_{1},\ldots ,h_{r})}(x|\lambda)\frac{t^{n}}{n!} \\& \quad = \sum_{n=0} ^{\infty} \sum _{m=0} ^{n} \lambda^{n-m}S_{1}(n,m){ \int _{\mathbb{Z}_{p}}}\cdots{ \int_{\mathbb{Z}_{p}}}q^{\sum_{i=1} ^{r} h_{i}y_{1}}(-x-y_{1} \\& \qquad {}- \cdots-y_{r})_{n}\,d\mu_{-q}(y_{1})\cdots \,d\mu_{-q}(y_{r})\frac {t^{n}}{n!} \\& \quad = { \int_{\mathbb{Z}_{p}}}\cdots{ \int_{\mathbb{Z}_{p}}}q^{\sum _{i=1} ^{r} h_{i}y_{i}}\sum_{n=0} ^{\infty}\binom{-x-y_{1}-\cdots -y_{r}}{n}\lambda^{-n}n! \\& \qquad {}\times\sum _{l=n} ^{\infty}S_{1}(l,n)\frac{(\lambda t)^{l}}{l!} \,d\mu_{-q}(y_{1})\cdots \,d\mu_{-q}(y_{r}) \\& \quad = { \int_{\mathbb{Z}_{p}}}\cdots{ \int_{\mathbb{Z}_{p}}}q^{\sum _{i=1} ^{r} h_{i}y_{i}}\sum_{n=0} ^{\infty}\binom{-x-y_{1}-\cdots -y_{r}}{n} \\& \qquad {}\times\lambda^{-n} \bigl(\log(1+\lambda t) \bigr)^{n} \,d\mu _{-q}(y_{1})\cdots \,d \mu_{-q}(y_{r}) \\& \quad = { \int_{\mathbb{Z}_{p}}}\cdots{ \int_{\mathbb{Z}_{p}}}q^{\sum _{i=1} ^{r} h_{i}y_{i}} \biggl(1+\frac{1}{\lambda}\log(1+ \lambda t) \biggr)^{-x-y_{1}-\cdots-y_{r}}\,d\mu_{-q}(y_{1})\cdots \,d \mu_{-q}(y_{r}) \\& \quad = \prod_{i=1} ^{r} \biggl( \frac{1+q}{q^{h_{i}+1}+1+\frac{1}{\lambda}\log (1+\lambda t)} \biggr) \biggl(1+\frac{1}{\lambda}\log(1+\lambda t) \biggr)^{r-x}. \end{aligned}$$
(36)
$$ \sum_{n=0} ^{\infty} {\widehat{ \operatorname{Ch}}}_{n,h,q} (x|\lambda)\frac {t^{n}}{n!}= \frac{1+q}{q^{h+1}+1+\frac{1}{\lambda}\log(1+\lambda t)} \biggl(1+\frac{1}{\lambda}\log(1+\lambda t) \biggr)^{1-x}. $$
(37)
By replacing t by \(\frac{1}{\lambda} (e^{\lambda t}-1 )\) in (36), we have
and, by (10), we get
By (38) and (39), we obtain the following theorem.
$$\begin{aligned} &\Biggl(\prod_{i=1} ^{r} \frac{[2]_{q}}{q^{h_{i}+1}+(1+t)} \Biggr) (1+t)^{r-x} \\ &\quad =\sum_{n=0} ^{\infty}\widehat{\operatorname{Ch}}_{n,q} ^{(h_{1},\ldots ,h_{r})}(x| \lambda)\frac{ (\frac{1}{\lambda}(e^{\lambda t}-1) )^{n}}{n!} \\ &\quad =\sum_{n=0} ^{\infty}\widehat{ \operatorname{Ch}}_{n,q} ^{(h_{1},\ldots ,h_{r})}(x|\lambda)\frac{1}{n!} \lambda^{-n}n!\sum_{l=n} ^{\infty} S_{2}(l,n)\frac{(\lambda t)^{l}}{l!} \\ &\quad =\sum_{n=0} ^{\infty} \Biggl(\sum _{m=0} ^{n} \widehat{\operatorname{Ch}}_{m,q} ^{(h_{1},\ldots,h_{r})}(x|\lambda)\lambda^{n-m}S_{2}(n,m) \Biggr) \frac{t^{n}}{n!}, \end{aligned}$$
(38)
$$\begin{aligned} & \Biggl(\prod_{i=1} ^{r} \frac{[2]_{q}}{q^{h_{i}+1}+(1+t)} \Biggr) (1+t)^{r-x} \\ &\quad = \Biggl(\prod_{i=1} ^{r-1} \Biggl( \sum_{n=0} ^{\infty} {\widehat { \operatorname{Ch}}}_{n,q} ^{(h_{i})}\frac{t^{n}}{n!} \Biggr) \Biggr) \Biggl(\sum_{n=0} ^{\infty} {\widehat{ \operatorname{Ch}}}_{n,q} ^{(h_{r})} (x)\frac{t^{n}}{n!} \Biggr) \\ &\quad =\sum_{n=0} ^{\infty} \biggl(\sum _{\substack{i_{1}+\cdots+i_{r}=n \\i_{1},\ldots,i_{r} \geq0}} \binom{n}{i_{1},\ldots,i_{r}}{\widehat { \operatorname{Ch}}}_{i_{1},q} ^{(h_{1})}\cdots{\widehat{ \operatorname{Ch}}}_{i_{r-1},q} ^{(h_{r-1})}{\widehat{\operatorname{Ch}}}_{i_{r},q} ^{(h_{r})}(x) \biggr)\frac{t^{n}}{n!}. \end{aligned}$$
(39)
Theorem 4.2
For
\(n \geq0\), we have
$$\begin{aligned}& \sum_{m=0} ^{n} \widehat{ \operatorname{Ch}}_{m,q} ^{(h_{1},\ldots,h_{r})}(x|\lambda ) \lambda^{n-m}S_{2}(n,m) \\& \quad =\sum_{\substack{i_{1}+\cdots+i_{r}=n\\ i_{1},\ldots,i_{r} \geq0}} \binom {n}{i_{1},\ldots,i_{r}}{\widehat{ \operatorname{Ch}}}_{i_{1},q} ^{(h_{1})}\cdots{\widehat { \operatorname{Ch}}}_{i_{r-1},q} ^{(h_{r-1})}{\widehat{\operatorname{Ch}}}_{i_{r},q} ^{(h_{r})}(x). \end{aligned}$$
Note that
and thus we see that
$$\begin{aligned}& \sum_{n=0} ^{\infty}{\widehat{ \operatorname{Ch}}}_{n,q} ^{(h_{1},\ldots ,h_{r})}(x|\lambda)\frac{t^{n}}{n!} \\& \quad = \prod_{i=1} ^{r} \biggl( \frac{1+q}{q^{h_{i}+1}+1+\frac{1}{\lambda}\log (1+\lambda t)} \biggr) \biggl(1+\frac{1}{\lambda}\log(1+\lambda t) \biggr)^{r-x} \\& \quad = \Biggl(\prod_{i=1} ^{r-1} \frac{(1+q)q^{-h_{i}-1}}{q^{-h_{i}-1} (1+\frac{1}{\lambda}\log(1+\lambda t) )+1} \Biggr) \biggl(1+\frac{1}{\lambda}\log(1+\lambda t) \biggr)^{r-1} \\& \qquad {} \times \biggl(\frac{(1+q)q^{-h_{r}-1}}{q^{-h_{r}-1} (1+\frac {1}{\lambda}\log(1+\lambda t) )+1} \biggr) \biggl(1+\frac {1}{\lambda} \log(1+\lambda t) \biggr)^{1-x} \\& \quad = \Biggl(\prod_{i=1} ^{r-1}q^{-h_{i}-1} \Biggr) \Biggl(\sum_{n=0} ^{\infty} \operatorname{Ch}_{n,q} ^{(-h_{1}-2,\ldots,-h_{r-1}-2)}(r-1|\lambda)\frac {t^{n}}{n!} \Biggr) \Biggl(\sum_{n=0} ^{\infty}{\widehat { \operatorname{Ch}}}_{n,-h_{r}-2,q}(x|\lambda)\frac{t^{n}}{n!} \Biggr) \\& \quad = \sum_{n=0} ^{\infty} \Biggl( \Biggl( \prod_{i=1} ^{r-1}q^{-h_{i}-1} \Biggr) \sum_{m=0} ^{\infty}\binom{n}{m} \operatorname{Ch}_{n-m,q} ^{(-h_{1}-2,\ldots,-h_{r-1}-2)}(r-1|\lambda){\widehat { \operatorname{Ch}}}_{m,-h_{r}-2,q}(x|\lambda) \Biggr)\frac{t^{n}}{n!}, \end{aligned}$$
$$\begin{aligned}& {\widehat{\operatorname{Ch}}}_{n,q} ^{(h_{1},\ldots,h_{r})}(x|\lambda) \\& \quad = \Biggl(\prod_{i=1} ^{r-1}q^{-h_{i}-1} \Biggr)\sum_{m=0} ^{\infty }\binom{n}{m} \operatorname{Ch}_{n-m,q} ^{(-h_{1}-2,\ldots,-h_{r-1}-2)}(r-1|\lambda ){\widehat{ \operatorname{Ch}}}_{m,-h_{r}-2,q}(x|\lambda). \end{aligned}$$
5 Conclusion
The Changhee polynomials were defined by Kim, and have been attempted the various generalizations by many researchers (see [1, 6, 12, 13, 24‐26, 28, 31]). The Changhee numbers (q-Changhee numbers, respectively) are closely relate with the Euler numbers (q-Euler numbers), the Stirling numbers of the first kind and second kind and the harmonic numbers, etc. which are interesting numbers of combinatorics, and pure and applied mathematics.
In this paper, we defined two types of the degenerate \((h,q)\)-Changhee polynomials and number, and found the relationship between the Stirling numbers of the first kind and second kind, \((h,q)\)-Euler numbers, q-Changhee numbers and those polynomials and numbers. It is a further problem to find the relationship between some special polynomials and degenerate \((h,q)\)-Changhee polynomials.
Acknowledgements
The authors would like to thank the referees for their valuable and detailed comments which have significantly improved the presentation of this paper.
Competing interests
The authors declare that they have no competing interests.
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