On the degenerate (h,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(h,q)$\end{document}-Changhee numbers and polynomials

The Changhee numbers and polynomials are introduced by Kim, Kim and Seo (Adv. Stud. Theor. Phys. 7(20):993–1003, 2013), and the generalizations of those polynomials are characterized. In this paper, we investigate a new q-analog of the higher order degenerate Changhee polynomials and numbers. We derive some new interesting identities related to the degenerate (h,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(h,q)$\end{document}-Changhee polynomials and numbers.


Introduction
For a fixed odd prime number p, we make use of the following notation. Z p , Q p , and 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 Q p , respectively. The p-adic norm is defined |p| p = p -1 (see [14,15,17,19,30]).
As is well known, the Stirling number of the first kind is defined by (x) n = x(x -1) · · · (xn + 1) = n l=0 S 1 (n, l)x l , ( 3 ) and the Stirling number of the second kind is given by the generating function to be e t -1 m = m! ∞ l=m S 2 (l, m) t l l! (see [3][4][5]11]).
By (3), we have The unsigned Stirling numbers of the first kind are given by [3,5]).
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 . 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.
Kim et al. introduced the Changhee polynomials of the first kind of order r, defined by the generating function to be t n n! (see [12,13]), and Moon et al. defined the q-Changhee polynomials of order r as follows: n,q (x) t n n! (see [28,31]).
By (2), we note that and thus we see that ∞ n=0 Ch (r) n (x) t n n! = Z p (1 + t) x+y dμ -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 Z p to be As is well known, the Euler polynomials are defined by the generating function to be [2,7,8,17,19,20,33,36,37]).
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.

q-Analog of degenerate Changhee polynomials
where h ∈ Z. By (14), we define the q-analog of degenerate Changhee polynomials by the generating function to be In the special case x = 0, Ch n,h,q (λ) = Ch n,h,q (0|λ) are called the q-analog of degenerate Changhee numbers. Note that and so we see that and, if we put h = 0, then lim λ→0 Ch (0) n,q (x|λ) = Ch n,q (x) and Ch n,0,q (x|λ) = Ch n,λ,q (x).
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 e xt (see [32]).

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 , . . . , h r ∈ Z and r ∈ N.

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, Ch By (7) and (34), it leads to × (x + y 1 + · · · + y r ) (m) dμ -q (y 1 ) · · · dμ -q (y r ) = n m=0 λ n-m S 1 (n, m) S 1 (m, l) (x + y 1 + · · · + y r ) l dμ -q (y 1 ) · · · dμ -q (y r ) Thus, we state the following theorem.

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.