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q-extensions of Genocchi numbers. (English) Zbl 1129.11008

The classical Genocchi numbers, G n are defined by means of the following generating function: ((2t)/(e t +1))=σ n=0 G n ((t n )/(n!)), where G 1 =1,G 3 =G 5 =G 7 ==0. Relations between Genocchi numbers, Bernoulli numbers and Euler polynomials are given by G n =(2-2 n+1 )B n =2nE 2n-1 (0). Genocchi numbers and polynomials are very important not only in Number Theory but also in the other areas in Mathematics and Mathematical Physics. The authors define q-Genocchi numbers and polynomials by means of the following generating functions, respectively:

F q (G) (t)=q(1+q)tσ n=0 (-1) n q n e [n]t =σ n=0 G n (q)((t n )/(n!)),


F q (G) (t)=F q (G) (q x t)e [x]t =σ n=0 G n (x,q)((t n )/(n!)),

where [x]=((1-q x )/(1-q)) and qC with |q|<1. The authors give interpolations functions of these numbers and polynomials at negative integers. They define p-adic q-l-function which interpolate q-Genocchi numbers at negative integers. They also give congruences for q-Genocchi numbers.

11B68Bernoulli and Euler numbers and polynomials
11S80Other analytic theory of local fields