New Bell–Sheffer Polynomial Sets

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.

Certain results on hybrid relatives of the Sheffer polynomials

“…The Sheffer polynomials are one of the most important class of polynomial sequences and have been extensively studied [2,10,11,13] not only due to the fact that they arise in numerous branches of mathematics but also because of their importance in applied sciences, such as physics and engineering. The Sheffer polynomials S n (p) for the pair (g(t), f (t)) are defined by the generating function [14,Pg.…”

On Sheffer polynomial families

“…In recent articles [1,2], new sets of Sheffer [3] and Brenke [4] polynomials, based on higher order Bell numbers [5][6][7][8][9], have been studied. Furthermore, several integer sequences [10] associated with the considered polynomials sets both of exponential [11,12] and logarithmic type have been introduced [2].…”

Differential Equations for Classical and Non-Classical Polynomial Sets: A Survey

“…This method has been recently applied in several articles (see [3][4][5][6][7][8][9]), which include works in collaboration with several authors. The most outstanding of them is Prof. Dr. Hari M. Srivastava, to whom this article is dedicated.…”