Existence and reduction of generalized apostol-bernoulli, apostol-euler and apostol-genocchi polynomials

Navas, L.M. ; Ruiz, F.J. (Universidad de Zaragoza) ; Varona, J.L.

Resumen: One can find in the mathematical literature many recent papers studying the generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, defined by means of generating functions. In this article we clarify the range of parameters in which these definitions are valid and when they provide essentially different families of polynomials. In particular, we show that, up to multiplicative constants, it is enough to take as the “main family” those given by \[ \Big ( \frac{2}{\lambda e^t+1} \Big )^\alpha e^{xt} = \sum _{n=0}^{\infty } \mathcal{E}^{(\alpha )}_{n}(x;\lambda ) \frac{t^n}{n!}\,, \qquad \lambda \in \mathbb{C}\setminus \lbrace -1\rbrace \,, \] and as an “exceptional family” \[ \Big ( \frac{t}{e^t-1} \Big )^\alpha e^{xt} = \sum _{n=0}^{\infty } \mathcal{B}^{(\alpha )}_{n}(x) \frac{t^n}{n!}\,, \] both of these for $\alpha \in \mathbb{C}$.

Idioma: Inglés

DOI: 10.5817/AM2019-3-157

Año: 2019

Publicado en: Archivum Mathematicum 55, 3 (2019), 157-165
ISSN: 0044-8753

Factor impacto SCIMAGO: 0.237 - Mathematics (miscellaneous) (Q3)

Financiación: info:eu-repo/grantAgreement/ES/DGA/E64
Financiación: info:eu-repo/grantAgreement/ES/MINECO/MTM2015-65888-C4-4-P

Tipo y forma: Article (Published version)

Área (Departamento): Área Análisis Matemático (Dpto. Matemáticas)