On the Generalized q-Genocchi Numbers and Polynomials of Higher-Order

where we use the technical method’s notation by replacing G x by Gn x , symbolically, see 1, 2 . In the special case x 0, Gn Gn 0 are called the nth Genocchi numbers. 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 Genocchi numbers for 2, 4, 6, . . . are −1, 1,−3, 17,−155, 2073, . . .. The first few prime Genocchi numbers are given by G6 −3 and G8 17. It is known that there are no other prime Genocchi numbers with n < 105. For a real or complex parameter α, the higher-order Genocchi polynomials are defined by ( 2t et 1 )α e ∞ ∑