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

Archivum Mathematicum (2019)

• Volume: 055, Issue: 3, page 157-165
• ISSN: 0044-8753

top

## Abstract

top
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 ${\left(\frac{2}{\lambda {e}^{t}+1}\right)}^{\alpha }{e}^{xt}=\sum _{n=0}^{\infty }{ℰ}_{n}^{\left(\alpha \right)}\left(x;\lambda \right)\frac{{t}^{n}}{n!}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2.0em}{0ex}}\lambda \in ℂ\setminus \left\{-1\right\}\phantom{\rule{0.166667em}{0ex}},$ and as an “exceptional family” ${\left(\frac{t}{{e}^{t}-1}\right)}^{\alpha }{e}^{xt}=\sum _{n=0}^{\infty }{ℬ}_{n}^{\left(\alpha \right)}\left(x\right)\frac{{t}^{n}}{n!}\phantom{\rule{0.166667em}{0ex}},$ both of these for $\alpha \in ℂ$.

## How to cite

top

Navas, Luis M., Ruiz, Francisco J., and Varona, Juan L.. "Existence and reduction of generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials." Archivum Mathematicum 055.3 (2019): 157-165. <http://eudml.org/doc/294715>.

@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 -

## References

top
1. Apostol, T., 10.2140/pjm.1951.1.161, Pacific J. Math. 1 (1951), 161–167, Pacific J. Math. 2 (1952), 10. (1951) MR0043843DOI10.2140/pjm.1951.1.161
2. Bayad, A., 10.1090/S0025-5718-2011-02476-2, Math. Comp. 80 (2011), 2219–2221. (2011) MR2813356DOI10.1090/S0025-5718-2011-02476-2
3. Hernández-Llanos, P., Quintana, Y., Urieles, A., 10.1007/s00025-014-0430-2, Results Math. 68 (2015), 203–225. (2015) MR3391500DOI10.1007/s00025-014-0430-2
4. Horadam, A.F., Genocchi polynomials, Applications of Fibonacci Numbers (G. E. Bergum, A. N. Philippou, Horadam, A. F., eds.), vol. 4, Kluwer, 1991, pp. 145–166. (1991) MR1193711
5. Kurt, B., 10.12691/tjant-1-1-11, Turkish Journal of Analysis and Number Theory 1 (2013), 54–58. (2013) DOI10.12691/tjant-1-1-11
6. Luo, Q.-M., 10.1090/S0025-5718-09-02230-3, Math. Comp. 78 (2009), 2193–2208. (2009) MR2521285DOI10.1090/S0025-5718-09-02230-3
7. Luo, Q.-M., Srivastava, H.M., 10.1016/j.jmaa.2005.01.020, J. Math. Anal. Appl. 308 (2005), 290–302. (2005) MR2142419DOI10.1016/j.jmaa.2005.01.020
8. Luo, Q.-M., Srivastava, H.M., Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind, Appl. Math. Comput. 217 (2011), 5702–5728. (2011) MR2770190
9. Navas, L.M., Ruiz, F.J., Varona, J.L., 10.1090/S0025-5718-2012-02568-3, Math. Comp. 81 (2012), 1707–1722. (2012) MR2904599DOI10.1090/S0025-5718-2012-02568-3
10. Nørlund, N.E., 10.1007/BF02401755, Acta Math. 43 (1922), 121–196. (1922) MR1555176DOI10.1007/BF02401755
11. Nørlund, N.E., Vorlesungen über Differenzenrechnung, 1st ed., Springer-Verlag, Berlin-Heidelberg, 1924. (1924) MR1549596
12. Srivastava, H.M., Choi, J., Zeta and $q$-zeta functions and associated series and integrals, Elsevier, 2012. (2012) MR3294573
13. Srivastava, H.M., Kurt, B., Simsek, Y., 10.1080/10652469.2012.690950, Integral Transforms Spec. Funct. 23 (2012), 939–940. (2012) MR2998907DOI10.1080/10652469.2012.690950

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.