Existence and reduction of generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials

@article{Navas2019,
abstract = {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 $\alpha \in \mathbb \{C\}$.},
author = {Navas, Luis M., Ruiz, Francisco J., Varona, Juan L.},
journal = {Archivum Mathematicum},
keywords = {Bernoulli polynomials; Nørlund polynomials; Apostol-Bernoulli polynomials; Apostol-Euler polynomials; Apostol-Genocchi polynomials; generating functions; Appell sequences},
language = {eng},
number = {3},
pages = {157-165},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Existence and reduction of generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials},
url = {http://eudml.org/doc/294715},
volume = {055},
year = {2019},
}

TY - JOUR
AU - Navas, Luis M.
AU - Ruiz, Francisco J.
AU - Varona, Juan L.
TI - Existence and reduction of generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials
JO - Archivum Mathematicum
PY - 2019
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 055
IS - 3
SP - 157
EP - 165
AB - 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 $\alpha \in \mathbb {C}$.
LA - eng
KW - Bernoulli polynomials; Nørlund polynomials; Apostol-Bernoulli polynomials; Apostol-Euler polynomials; Apostol-Genocchi polynomials; generating functions; Appell sequences
UR - http://eudml.org/doc/294715
ER -