Differential Equation and Recursive Formulas of Sheffer Polynomial Sequences

Abstract

We derive a differential equation and recursive formulas of Sheffer polynomial sequences utilizing matrix algebra. These formulas provide the defining characteristics of, and the means to compute, the Sheffer polynomial sequences. The tools we use are well-known Pascal functional and Wronskian matrices. The properties and the relationship between the two matrices simplify the complexity of the generating functions of Sheffer polynomial sequences. This work extends He and Ricci's work (2002) to a broader class of polynomial sequences, from Appell to Sheffer, using a different method. The work is self-contained.

A matrix approach to some identities involving Sheffer polynomial sequences

“…From the method of Youn and Yang in [17], we derive some identities of special polynomials satisfied by Sheffer polynomial sequences in this section. …”

“…The following proposition, stated as Property 1 in [17], is of fundamental use throughout this paper. P n ahðtÞ þ blðtÞ ½ ¼ aP n hðtÞ ½ þbP n lðtÞ ½ ; W n ahðtÞ þ blðtÞ ½ ¼ aW n hðtÞ ½ þbW n lðtÞ ½ :…”

“…In this past few decades, there has been a renewed interest in Sheffer polynomials (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]). …”

“…; s n x ð Þ ½ . A simple but elegant method was adopted in [17] in order to derive a differential equation and recursive formulas for Sheffer polynomial sequences. Namely, they used the so called the generalized Pascal functional matrix of an analytic function and the Wronskian matrix of several analytic functions.…”

A linear algebra approach to the hybrid Sheffer–Appell polynomials

“…It is important to note that in the expressions  n [h(x, t)] t=0 and  n [h(x, t)] t=0 , we consider t as the working variable and x as a parameter. We recall certain important properties and relationships between the Pascal functional and Wronskian matrices [14].…”