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On the \(q\)-extension of Euler and Genocchi numbers. (English) Zbl 1112.11012
The author considers a new \(q\)-extension of ordinary Euler numbers and polynomials, and also a new \(q\)-extension of Genocchi numbers and polynomials. They are defined by using the generating functions as follows: \[ \begin{aligned} \sum^\infty_{n=0} E_{n,q}{t^n\over n!} &= [2]_q e^{{t\over 1-q}} \sum^\infty_{j=0} {(-1)^j\over 1+ q^{j+1}}\Biggl({1\over 1-q}\Biggr)^j {t^j\over j!},\\ \sum^\infty_{n=0} E_{n,q}(x){t^n\over n!} &= [2]_q e^{{t\over 1-q}}\sum^\infty_{j=0} {(-1)^j q^{jx}\over 1+ q^{j+1}} \Biggl({1\over 1-q}\Biggr)^j{t^j\over j!},\\ \sum^\infty_{n=0} G_{n,q}{t^n\over n!} &= [2]_q t\sum^\infty_{n=0} (-1)^n q^n e^{[n]_q t}\text{ and }\\ \sum^\infty_{n=0} G_{n,q}(x){t^n\over n!} &= [2]_q t\sum^\infty_{n=0} (-1)^n q^{n+x} e^{[n+ x]_q t},\end{aligned} \] where \([n]_q= 1+ q+\cdots+ q^{n-1}\) for a positive integer \(n\). He obtains several identities for these \(q\)-extensions including the \(q\)-analogues of the formulae \[ E_m(x)= \sum^m_{k=0} {m\choose k}{G_{k+1}\over k+1} x^{m-k}\quad\text{and}\quad (n^m- n)G_m= \sum^{m-1}_{k-1} {m\choose k} n^k G_k Z_{m-k}(n- 1) \] for ordinary Euler polynomials \(E_m(x)\) and Genocchi numbers \(G_m\), where \(Z_m(n)= 1^m- 2^m+\cdots+ (-1)^{n+1} n^m\).
Reviewer: Kaori Ota (Tokyo)

MSC:
11B68 Bernoulli and Euler numbers and polynomials
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