Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1129.11008
Cenkci, Mehmet; Can, Mümün; Kurt, Veli
$q$-extensions of Genocchi numbers. (English)
[J] J. Korean Math. Soc. 43, No. 1, 183-198 (2006). ISSN 0304-9914

The classical Genocchi numbers, $G_{n}$ are defined by means of the following generating function: $((2t)/(e^{t}+1))=\sigma_{n=0}^{\infty}G_{n}((t^{n})/(n!))$, where $G_{1}=1, G_{3}=G_{5} =G_{7}= \dots =0$. Relations between Genocchi numbers, Bernoulli numbers and Euler polynomials are given by $G_{n} = (2-2^{n + 1})B_{n} = 2nE_{2n-1}(0)$. Genocchi numbers and polynomials are very important not only in Number Theory but also in the other areas in Mathematics and Mathematical Physics. The authors define $q$-Genocchi numbers and polynomials by means of the following generating functions, respectively: $$F_{q}^{(G)}(t)=q(1+q)t\sigma_{n=0}^{\infty}(-1)^{n}q^{n}e^{[n]t}=\sigma_{n=0}^{\infty}G_{n}(q)((t^{n})/(n!)), $$ and $$F_{q}^{(G)}(t)=F_{q}^{(G)}(q^{x}t)e^{[x]t}=\sigma_{n=0}^{\infty}G_{n}(x,q)((t^{n})/(n!)),$$ where $[x]=((1-q^{x})/(1-q)) $ and $q\in C$ with $\vert q\vert<1$. The authors give interpolations functions of these numbers and polynomials at negative integers. They define $p$-adic $q$-$l$-function which interpolate $q$-Genocchi numbers at negative integers. They also give congruences for $q$-Genocchi numbers.
[Yilmaz Simsek (Antalya)]

MSC 2000:: *11B68 Bernoulli numbers, etc.
11S80 Other analytic theory of local fields

Keywords: Bernoulli numbers; Genocchi numbers and polynomials; $q$-Genocchi numbers and polynomials

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]