Differential Equation and Recursive Formulas of Sheffer Polynomial Sequences

Heekyung Youn,Yongzhi Yang

doi:10.5402/2011/476462

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.

Highlights

Sheffer polynomial sequences arise in numerous problems of applied mathematics, theoretical physics, approximation theory, and several other mathematical branches
In the past few decades, there has been a renewed interest in Sheffer polynomials. di Bucchianico recently summarized and documented more than five hundred old and new findings related to the study of Sheffer polynomial sequences in 1
In 2, He and Ricci developed the differential equation and recursive formula for Appell polynomials, which is a subclass of Sheffer polynomial sequences

Summary

Introduction

Sheffer polynomial sequences arise in numerous problems of applied mathematics, theoretical physics, approximation theory, and several other mathematical branches. Di Bucchianico recently summarized and documented more than five hundred old and new findings related to the study of Sheffer polynomial sequences in 1. One aspect of such study is to find a differential equation and recursive formulas for Sheffer polynomial sequences. In 2 , He and Ricci developed the differential equation and recursive formula for Appell polynomials, which is a subclass of Sheffer polynomial sequences. We derive differential equation and recursive formulas for Sheffer polynomial sequences by using matrix algebra. An example will illustrate how these three forms of recursive formulas are useful in their own right

Preliminaries

Sheffer Polynomial Sequence and Its Differential Equation

Recurrence Relations for the Sheffer Polynomials

The First Recursive Formula for the Sheffer Polynomials

The Second Recursive Formula for the Sheffer Polynomials

The Third Recursive Formula for the Sheffer Polynomials