The dual of number sequences, Riordan polynomials, and Sheffer polynomials

In this paper we introduce different families of numerical and polynomial sequences by using Riordan pseudo involutions and Sheffer polynomial sequences. Many examples are given including dual of Hermite numbers and polynomials, dual of Bell numbers and polynomials, among other. The coefficients of some of these polynomials are related to the counting of different families of set partitions and permutations. We also studied the dual of Catalan numbers and dual of Fuss-Catalan numbers, giving several combinatorial identities.

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

By using the monomiality principle and general results on Sheffer polynomial sets, the differential equation satisfied by several old and new polynomial sets is shown.

On Sheffer polynomial families

Attention is focused to particular families of Sheffer polynomials which are different from the classical ones because they satisfy non-standard differential equations, including some of fractional type. In particular Sheffer polynomial families are considered whose characteristic elements are based on powers or exponential functions.

New Bell–Sheffer Polynomial Sets

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.

On multivariable cumulant polynomial sequences with applications

A new family of polynomials, called cumulant polynomial sequence, and its extension to the multivariate case is introduced relied on a purely symbolic combinatorial method. The coefficients are cumulants, but depending on what is plugged in the indeterminates, moment se- quences can be recovered as well. The main tool is a formal generalization of random sums, when a not necessarily integer-valued multivariate random index is considered. Applications are given within parameter estimations, L\'evy processes and random matrices and, more generally, problems involving multivariate functions. The connection between exponential models and multivariable Sheffer polynomial sequences offers a different viewpoint in employing the method. Some open problems end the paper.

An algebraic exposition of umbral calculus with application to general linear interpolation problem: A survey

A systematic exposition of Sheffer polynomial sequences via determinantal form is given. A general linear interpolation problem related to Sheffer sequences is considered. It generalizes many known cases of linear interpolation. Numerical examples and conclusions close the paper.