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On Genocchi numbers and polynomials. (English) Zbl 1217.11024

The author calls f a uniformly differential function at a p (fUD( p ) for short) if the difference quotients, F f (x,y)=(f(x)-f(y))/(x-y), have a limit f ' (a) as (x,y)(a,a). Denote as usual [x] -q =1-(-q) x 1+q, [x] q =1-q x 1-q. Then for fUD( p ), the fermionic p-adic invariant q-integral on p is defined as

I -q (f)= p f(x)dμ -q (x)=lim N 1 [p N ] -q x=0 p N -1 f(x)(-q) x ·

Especially, one has

I -1 (f)=lim q1 I -q (f)= p f(x)dμ -1 (x)·

In the paper under review, the authors study certain integral equations related to I -q (f) from which they obtain several properties of Genocchi numbers and polynomials.

The main purpose is to derive the distribution relations of the Genocchi polynomials, and to construct the Genocchi zeta function which interpolates the Genocchi polynomials at negative integers.

11B68Bernoulli and Euler numbers and polynomials
05A10Combinatorial functions
11S40Zeta functions and L-functions of local number fields