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

The classical Genocchi numbers, ${G}_{n}$ are defined by means of the following generating function: $\left(\left(2t\right)/\left({e}^{t}+1\right)\right)={\sigma }_{n=0}^{\infty }{G}_{n}\left(\left({t}^{n}\right)/\left(n!\right)\right)$, where ${G}_{1}=1,{G}_{3}={G}_{5}={G}_{7}=\cdots =0$. Relations between Genocchi numbers, Bernoulli numbers and Euler polynomials are given by ${G}_{n}=\left(2-{2}^{n+1}\right){B}_{n}=2n{E}_{2n-1}\left(0\right)$. 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:

${F}_{q}^{\left(G\right)}\left(t\right)=q\left(1+q\right)t{\sigma }_{n=0}^{\infty }{\left(-1\right)}^{n}{q}^{n}{e}^{\left[n\right]t}={\sigma }_{n=0}^{\infty }{G}_{n}\left(q\right)\left(\left({t}^{n}\right)/\left(n!\right)\right),$

and

${F}_{q}^{\left(G\right)}\left(t\right)={F}_{q}^{\left(G\right)}\left({q}^{x}t\right){e}^{\left[x\right]t}={\sigma }_{n=0}^{\infty }{G}_{n}\left(x,q\right)\left(\left({t}^{n}\right)/\left(n!\right)\right),$

where $\left[x\right]=\left(\left(1-{q}^{x}\right)/\left(1-q\right)\right)$ 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 of local fields