On the Modified q-Euler Polynomials with Weight and Weak Weights

In this paper, we construct a new q-extension of Euler numbers and polynomials with weight related to fermionic p-adic q-integral on Zp and give new explicit formulas related to these numbers and polynomials.


Introduction
Throughout this paper Z p , Q p and C p will respectively denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of Q p . Let ν p be the normalized exponential valuation of C p with |p| p = p −νp(p) = 1 p . In this paper, we assume that q ∈ C p with |1 − q| p < p − 1 p−1 . So that q x = exp (x log q) for x ∈ Z p . The q-number of x is denoted by [x] q = 1−q x 1−q . Note that lim q→1 [x] q = x. Let d be a fixed integer bigger than 0 and let p be a fixed prime number and (d, p) = 1. We set (a + dpZ p ) , a + dp N Z p = x ∈ X|x ≡ a (mod dp N ) , where a ∈ Z lies in 0 ≤ a < dp N , (see [2][3][4][5][6][7][8][9][10][11]).
Let C(Z p ) be the space of continuous functions on Z p . For f ∈ C(Z p ), the fermionic p-adic q-integral on Z p is defined by Kim by [7,8,9]).
with the usual convention about replacing E n (x) by E n (x). In the special case, x = 0, E n (0) = E n are called the n-th Euler numbers. In [7,8], T. Kim defined the q-Euler numbers as follows: with the usual convection of replacing E n by E n,q . From (1.2), we also derive [7,11]). By using an invariant p-adic q-integral on Z p , a q-extension of ordinary Euler polynomials which are called q-Euler polynomials are considered and investigated by Kim [4,5,6,7,9]. For x ∈ Z p , q-Euler polynomials are defined as follows: By (1.3), the following relation is hold: Recently, Ryoo considered the weighted q-Euler polynomials which are a slightly different Kim's weighted q-Euler polynomials as follows: n,q (x) = p [x + y] n q α dμ −q (y), for n ∈ N and α ∈ Z, (see [12]).
In the special case, n,q are called the n-th q-Euler numbers with weight α, and showed that (1.4) And in [10], C. S. Ryoo defied q-Euler polynomials n,q (x) as follows: In the special case, x = 0, n,q are called the n-th q-Euler numbers with weak weight α, and show that A systemic study of some families of the modified q-Euler polynomials with weight is presented by using the multivariate fermionic p-adic integral on Z p . The study of these higher-order q-Euler numbers and polynomials yields an interesting q-analogue of identities for Stirling numbers.
In recent years, many mathematicians and physicists have investigated zeta functions, multiple zeta functions, L-functions, and multiple q-Bernoulli numbers and polynomials, mainly because of their interest and importance. These functions and polynomials are used not only in complex analysis and mathematical physics, but also in p-adic analysis and other areas. In particular, multiple zeta functions and multiple L-functions occur within the context of knot theory, quantum field theory, applied analysis, and number theory (see [1][2][3][4][5][6][7][8][9][10][11][12][13]).
In this paper, we construct new q-extension of Euler numbers and polynomials with weight α and weak weight β related to fermionic p-adic q-integral on Z p , and give new explicit formulas related to these numbers and polynomials. Also, we give another definition of the Euler polynomials of higher-order with weight α and weak weight β 1 , . . . , β k .

A new approach of q-Euler polynomials with weight α and weak weight β
Consider the q-Euler number with weight α and weak weight β E (α,β) n,q . Note that, by (1.1), As a new q-extension of Euler polynomials, we define the modified q-Euler polynomials with weight α and weak weight βẼ ij which are defined by fermionic From (2.1) and (2.2), we obtain the following equation:

Corollary 2.3.
For n ≥ 0, we havẽ where E n,q are the n-th q-Euler numbers with weight α and weak weight β.

3.
Multiple of modified q-Euler polynomials with weight α and weak weights β 1 , . . . , β k Let us consider the following multiple q-Euler number with weight α and weak weight β 1 , . . . , β k as follow; . Thus, we define the multiple modified q-Euler polynomials with weight α and weak weight β 1 , . . . , β k as follow; .

Consider the equation
Since and Thus, we have the following result.