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On Genocchi numbers and polynomials. (English) Zbl 1217.11024
The author calls $f$ a uniformly differential function at $a\in\Bbb Z_p$ ($f\in\text{UD}(\Bbb Z_p)$ 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 $f\in\text{UD}(\Bbb Z_p)$, the fermionic $p$-adic invariant $q$-integral on $\Bbb Z_p$ is defined as $$I_{-q}(f)=\int_{\Bbb Z_p}f(x)\,d\mu_{-q}(x)=\lim_{N\to\infty}\frac 1{[p^N]_{-q}} \sum_{x=0}^{p^N-1}f(x)(-q)^x.$$ Especially, one has $$I_{-1}(f)=\lim_{q\to 1}I_{-q}(f)=\int_{\Bbb Z_p}f(x)\,d\mu_{-1}(x).$$ 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:
11B68Bernoulli and Euler numbers and polynomials
05A10Combinatorial functions
11S40Zeta functions and $L$-functions of local number fields
WorldCat.org
Full Text: DOI EuDML
References:
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