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Zbl 1217.11024
Rim, Seog-Hoon; Park, Kyoung Ho; Moon, Eun Jung
On Genocchi numbers and polynomials.
(English)
[J] Abstr. Appl. Anal. 2008, Article ID 898471, 7 p. (2008). ISSN 1085-3375; ISSN 1687-0409/e

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.
[Olaf Ninnemann (Berlin)]
MSC 2000:
*11B68 Bernoulli numbers, etc.
05A10 Combinatorial functions
11S40 Zeta functions and L-functions of local number fields

Keywords: fermionic $p$-adic invariant $q$-integral; Genocchi numbers; Genocchi polynomials; Dirichlet-type Genocchi $l$-function; value of Genocchi zeta function at positive integers; Hurwitz-type Genocchi zeta function

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