## $$q$$-extensions of Genocchi numbers.(English)Zbl 1129.11008

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 $$| q|<1$$. 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.

### MSC:

 11B68 Bernoulli and Euler numbers and polynomials 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.)
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