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Axioms 2018, 7(4), 71; https://doi.org/10.3390/axioms7040071

New Bell–Sheffer Polynomial Sets

1
Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Largo San Leonardo Murialdo, 1, 00146 Roma, Italy
2
Sezione di Matematica, International Telematic University UniNettuno, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy
*
Author to whom correspondence should be addressed.
Received: 20 July 2018 / Accepted: 2 October 2018 / Published: 8 October 2018
(This article belongs to the Special Issue Mathematical Analysis and Applications)
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Abstract

In recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers, and several integer sequences related to them, have been studied. The method used in previous articles, and even in the present one, traces back to preceding results by Dattoli and Ben Cheikh on the monomiality principle, showing the possibility to derive explicitly the main properties of Sheffer polynomial families starting from the basic elements of their generating functions. The introduction of iterated exponential and logarithmic functions allows to construct new sets of Bell–Sheffer polynomials which exhibit an iterative character of the obtained shift operators and differential equations. In this context, it is possible, for every integer r, to define polynomials of higher type, which are linked to the higher order Bell-exponential and logarithmic numbers introduced in preceding papers. Connections with integer sequences appearing in Combinatorial analysis are also mentioned. Naturally, the considered technique can also be used in similar frameworks, where the iteration of exponential and logarithmic functions appear. View Full-Text
Keywords: Sheffer polynomials; generating functions; monomiality principle; shift operators; combinatorial analysis Sheffer polynomials; generating functions; monomiality principle; shift operators; combinatorial analysis
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 (CC BY 4.0).
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Natalini, P.; Ricci, P.E. New Bell–Sheffer Polynomial Sets. Axioms 2018, 7, 71.

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