On the families of \(q\)-Euler polynomials and their applications. (English) Zbl 1311.05010
Summary: In this paper, we focus on applications of \(q\)-Euler polynomials and obtain some new combinatorial relations by using \(q\)-adic \(q\)-integral on \(\mathbb{Z}_p\). Moreover, we derive distribution formula (multiplication theorem) for Dirichlet type of \(q\)-Euler numbers and polynomials with weight \(\alpha\). Also we apply the method of analytic continuation of \(q\)-Euler polynomials which is the main result of this paper.
MSC:
| 05A10 | Factorials, binomial coefficients, combinatorial functions |
| 05A30 | \(q\)-calculus and related topics |
| 11B65 | Binomial coefficients; factorials; \(q\)-identities |
| 11B68 | Bernoulli and Euler numbers and polynomials |
| 11B73 | Bell and Stirling numbers |
Keywords:
Euler numbers and polynomials; \(q\)-Euler numbers and polynomials; weighted \(q\)-Euler numbers and polynomials; weighted \(q\)-Euler-zeta function; \(p\)-adic \(q\)-integral on \(\mathbb{Z}_p\)References:
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