Abstract
We discuss new concept of the -extension of Genocchi numbers and give some relations between -Genocchi polynomials and -Euler numbers.
Highlights
The Genocchi numbers Gn, n = 0,1,2, . . . , which can be defined by the generating function
It is easy to find the values G1 = 1, G3 = G5 = G7 = · · · = 0, and even coefficients are given by G2m = 2(1 − 22n)B2n = 2nE2n−1(0), where Bn is a Bernoulli number and En(x) is an Euler polynomial
In the previous paper [4, 9], the author constructed the q-extension of Euler polynomials by using p-adic q-fermionic integral on Zp as follows: En,q(x) = [t + x]nqdμ−q(t), Zp where μ−q x + pN Zp
Summary
We discuss new concept of the q-extension of Genocchi numbers and give some relations between q-Genocchi polynomials and q-Euler numbers. In the previous paper [4, 9], the author constructed the q-extension of Euler polynomials by using p-adic q-fermionic integral on Zp as follows: En,q(x) = [t + x]nqdμ−q(t), Zp where μ−q x + pN Zp The q-extension of Genocchi numbers is defined as gq∗(t) We derive the generating function of ξn,q as follows: Fq(t)
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