Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

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

Highlights
Master Server