Abstract and Applied Analysis
Volume 2004 (2004), Issue 7, Pages 613-623

Generalizations of the Bernoulli and Appell polynomials

Gabriella Bretti,1 Pierpaolo Natalini,2 and Paolo E. Ricci3

1Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università degli Studi di Roma “La Sapienza”, Roma 00161, Italy
2Dipartimento di Matematica, Università degli Studi Roma Tre, Roma 00146, Italy
3Dipartimento di Matematica, Istituto “Guido Castelnuovo”, Università degli Studi di Roma “La Sapienza”, Roma 00185, Italy

Received 19 July 2002

Copyright © 2004 Gabriella Bretti et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We first introduce a generalization of the Bernoulli polynomials, and consequently of the Bernoulli numbers, starting from suitable generating functions related to a class of Mittag-Leffler functions. Furthermore, multidimensional extensions of the Bernoulli and Appell polynomials are derived generalizing the relevant generating functions, and using the Hermite-Kampé de Fériet (or Gould-Hopper) polynomials. The main properties of these polynomial sets are shown. In particular, the differential equations can be constructed by means of the factorization method.