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On Genocchi numbers and polynomials. (English) Zbl 1217.11024
The author calls \(f\) a uniformly differential function at \(a\in\mathbb Z_p\) (\(f\in\text{UD}(\mathbb Z_p)\) for short) if the difference quotients, \(F_f(x,y)=(f(x)-f(y))/(x-y)\), have a limit \(f'(a)\) as \((x,y)\to (a,a)\). Denote as usual \([x]_{-q}=\frac{1-(-q)^x}{1+q}\), \([x]_q={1-q^x}{1-q}\). Then for \(f\in\text{UD}(\mathbb Z_p)\), the fermionic \(p\)-adic invariant \(q\)-integral on \(\mathbb Z_p\) is defined as
\[ I_{-q}(f)=\int_{\mathbb Z_p}f(x)\,d\mu_{-q}(x)=\lim_{N\to\infty}\frac 1{[p^N]_{-q}} \sum_{x=0}^{p^N-1}f(x)(-q)^x. \]
Especially, one has \[ I_{-1}(f)=\lim_{q\to 1}I_{-q}(f)=\int_{\mathbb Z_p}f(x)\,d\mu_{-1}(x). \] 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.

MSC:
11B68 Bernoulli and Euler numbers and polynomials
05A10 Factorials, binomial coefficients, combinatorial functions
11S40 Zeta functions and \(L\)-functions
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