On degenerate q-Euler polynomials

In this paper, we consider degenerate Carlitz's type q-Euler polynmials and numbers and we investigate some identities arising from the fermionic p-adic integral equations and the generating function of thoe polynomials.


Introduction
Let p be a fixed odd prime number. Throughout this paper, Z p , Q p and C p will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers and the completion of the algebraic closure of Q p . Let ν p be the normalized exponential valuation of C p with |p| p = p −νp(p) = 1 p . Let q be an indeterminate in C p such that |q − 1| p < p − 1 p−1 so that q x = exp(x log q). The q-analogue of x is defined as [x] q = 1−q x 1−q . Note that lim q→1 = x. As is well known, the Euler polynomials are defined by the generating function to be 2 e t + 1 e xt = ∞ n=0 E n (x) t n n! , (see [1][2][3][4][5][6][7][8][9][10][11]).
From (1.5), we have The q-Euler polynomials can be represented by the fermionic p-adic q-integral on Z p as follows: ∞ n=0 E n,q (x) t n n! = Zp e [x+y]qt dµ −q (y), (see [5,6] In this paper, we consider the degenerate Carlitz q-Euler polynomials and numbers which are derived from the fermionic p-adic q-integral on Z p and we investigate some properties and identities of those polynomials.

Degenerate Carlitz q-Bernoulli numbers and polynomials
In this section, we assume that From (1.6) and (2.1), we consider the degenerate q-Euler polynomials which are given by the generating function to be (2.5) By comparing the coefficients on the both sides of (2.5), we obtain the following theorem.
where S 1 (n, m) is the Stirling number of the first kind.
Replacing t by 1 On the other hand, (2.7) Therefore, by (2.6) and (2.7), we obtain the following theorem. where S 2 (n, m) is the Stirling number of the second kind.
From (1.7), we can easily derive the following equation: (2.10) Thus, by (2.10), we get the generating function of the q-Euler polynomials as follows: From (2.2) and (2.11), we can derive the generating function of degenerate q-Euler polynomials which is given by (2.12) (2.13) Therefore, by (2.13), we obtain the following theorem.
It is not difficult to show that Thus, by Theorem 2.5 and (2.15), we obtain the following theorem.
Let r ∈ N. Now, we recall the Carlitz's q-Euler polynomials of order r which are given by the generating function to be Thus, by (2.16), we get n,q (x), (n ≥ 0), (see [5]).
(2.17) From (2.17), we have (2.18) Thus, by (2.18), we get the generating function of Carlitz's q-Euler polynomials of order r which given by We consider degenerate q-Euler polynomials of order r which are given by the generating function to be (2.21) By comparing the coefficients on the both sides of (2.21), we obtain the following theorem.