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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: \(((2t)/(e^{t}+1))=\sigma_{n=0}^{\infty}G_{n}((t^{n})/(n!))\), 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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