Symmetric Properties of Carlitz’s Type q -Changhee Polynomials

: Changhee polynomials were introduced by Kim, and the generalizations of these polynomials have been characterized. In our paper, we investigate various interesting symmetric identities for Carlitz’s type q -Changhee polynomials under the symmetry group of order n arising from the fermionic p -adic q -integral on Z p .


Introduction
For an odd prime number p, Z p , Q p , and C p denote the ring of p-adic integers, the field of p-adic rational numbers, and the completions of algebraic closure of Q p , respectively, throughout this paper.
In current paper, we construct symmetric identities for the Carlitz's type q-Changhee polynomials under the symmetry group of order n arising from the fermionic p-adic q-integral on Z p , and the proof methods which was used in the Kim's previous researches are also used as good tools in this paper (see [5,10,14,16,[32][33][34][35][36][37][38][39]).

Symmetric Identities for the Carlitz's Type q-Changhee Polynomials
Let t ∈ C p with |t| p < p − 1 p−1 , and let S n be the symmetry group of degree n. For positive integers w 1 , w 2 , . . . , w n with w i ≡ 1 (mod 2) for each i = 1, 2, . . . n, we consider the following integral equation for the fermionic p-adic q-integral on Z p ; From (13), we get If we put then, by (14), we know that F(w 1 , w 2 , . . . , w n ) is invariant for any permutation σ ∈ S n . Hence, by (14) and (15), we obtain the following theorem.
Ch m,q w 1 w 2 ···w n−1 w n x + w n for each positive integer n. Thus, by Theorem 1 and (17), we obtain the following corollary.

Conclusion
The Changhee numbers are closely related with the Euler numbers, the Stirling numbers of the first kind and second kind and the harmonic numbers, and so on. Throughout this paper, we investigate that the function F(w σ (1) , w σ (2) , . . . , w σ(n) ) for the Carlitz's type q-Changhee polynomials is invariant under the symmetry group σ ∈ S n . From the invariance of F(w σ (1) , w σ (2) , . . . , w σ(n) ), σ ∈ S n , we construct symmetric identities of the Carlitz's type q-Changhee polynomials from the fermionic p-adic q-integral on Z p . As Bernoulli and Euler polynomials, our properties on the Carlitz's type q-Changhee polynomials play an crucial role in finding identities for numbers in algebraic number theory.
Author Contributions: All authors contributed equally to this work; All authors read and approved the final manuscript.