Abstract
We first consider the -extension of the generating function for the higher-order generalized Genocchi numbers and polynomials attached to . The purpose of this paper is to present a systemic study of some families of higher-order generalized -Genocchi numbers and polynomials attached to by using the generating function of those numbers and polynomials.
Highlights
As a well known definition, the Genocchi polynomials are defined by2t et 1 ext eG x t ∞ Gn n0 x tn n!|t| < π, 1.1 where we use the technical method’s notation by replacing Gn x by Gn x, symbolically, see 1, 2
From the definition of Genocchi numbers, we note that G1 1, G3 G5 G7 · · · 0, and even coefficients are given by G2n 2 1 − 22n B2n 2nE2n−1 0 see 3, where Bn is a Bernoulli number and En x is an Euler polynomial
The first few prime Genocchi numbers are given by G6 −3 and G8 17
Summary
As a well known definition, the Genocchi polynomials are defined by2t et 1 ext eG x t ∞ Gn n0 x tn n!|t| < π, 1.1 where we use the technical method’s notation by replacing Gn x by Gn x , symbolically, see 1, 2. In the special case x 0, Gn Gn 0 are called the nth Genocchi numbers. The first few Genocchi numbers for 2, 4, 6, . It is known that there are no other prime Genocchi numbers with n < 105.
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