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.
Highlights
In recent articles [1,2], new sets of Sheffer [3] and Brenke [4] polynomials, based on higher orderBell numbers [2,5,6,7], have been studied
Several integer sequences associated [8] with the considered polynomials sets both of exponential [9,10] and logarithmic type have been introduced [1]
It is worth noting that exponential and logarithmic polynomials have been recently studied in the multidimensional case [11,12,13]
Summary
In recent articles [1,2], new sets of Sheffer [3] and Brenke [4] polynomials, based on higher order. Several integer sequences associated [8] with the considered polynomials sets both of exponential [9,10] and logarithmic type have been introduced [1]. It is worth noting that exponential and logarithmic polynomials have been recently studied in the multidimensional case [11,12,13]. New sets of Bell–Sheffer polynomials are considered and some particular cases are analyzed. It is worth noting that the Sheffer A-type 0 polynomial sets have been approached with elementary methods of linear algebra (see, e.g., [14,15,16] and the references therein). Connection with umbral calculus has been recently emphasized (see, e.g., [17,18] and the references therein)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
