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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.

\[ I_{-q}(f)=\int_{\mathbb Z_p}f(x)\,d\mu_{-q}(x)=\lim_{N\to\infty}\frac 1{[p^N]_{-q}} \sum_{x=0}^{p^N-1}f(x)(-q)^x. \]

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.

Reviewer: Olaf Ninnemann (Berlin)

### MSC:

11B68 | Bernoulli and Euler numbers and polynomials |

05A10 | Factorials, binomial coefficients, combinatorial functions |

11S40 | Zeta functions and \(L\)-functions |

### 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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\textit{S.-H. Rim} et al., Abstr. Appl. Anal. 2008, Article ID 898471, 7 p. (2008; Zbl 1217.11024)

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