About this Journal Submit a Manuscript Table of Contents
ISRN Discrete Mathematics
Volume 2011 (2011), Article ID 476462, 16 pages
http://dx.doi.org/10.5402/2011/476462
Research Article

Differential Equation and Recursive Formulas of Sheffer Polynomial Sequences

Department of Mathematics, University of St. Thomas, 2115 Summit Avenue, Saint Paul, MN 55105-1079, USA

Received 3 August 2011; Accepted 7 September 2011

Academic Editors: M. Chlebík, K. Eriksson, and M. C. Wilson

Copyright © 2011 Heekyung Youn and Yongzhi Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

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]. One aspect of such study is to find a differential equation and recursive formulas for Sheffer polynomial sequences. For instance, in [2], He and Ricci developed the differential equation and recursive formula for Appell polynomials, which is a subclass of Sheffer polynomial sequences. In this paper, we derive differential equation and recursive formulas for Sheffer polynomial sequences by using matrix algebra.

The remainder of the paper is organized as follows. In Section 2, we define Pascal functional and Wronskian matrices for analytic functions and derive their properties and the relationships between the functional matrices. In Section 3, we develop a differential equation for Sheffer polynomial sequences and present differential equations for some well-known Sheffer polynomials such as Laguerre, Lower factorial, Exponential, and Hermite polynomials. In Section 4, we discuss three recursive formulas for Sheffer polynomial sequences. An example will illustrate how these three forms of recursive formulas are useful in their own right.

2. Preliminaries

Let us start with the definitions of the generalized Pascal functional matrix of an analytic function [3] and the Wronskian matrix of several analytic functions. To avoid any unnecessary confusion, we use 𝑓(𝑘) to stand for the 𝑘th order derivative of 𝑓 and use 𝑓𝑘 to represent the 𝑘th power of 𝑓 in the entire paper. In addition, 𝑓(0)=𝑓 and 𝑓0=1.

Definition 2.1. Let 𝑓(𝑡) be an analytic function. The generalized Pascal functional matrix of 𝑓(𝑡), denoted by 𝒫𝑛[𝑓(𝑡)], is an (𝑛+1) by (𝑛+1) matrix and is defined as 𝒫𝑛[]𝑓(𝑡)𝑖,𝑗=𝑖𝑗𝑓(𝑖𝑗)(𝑡)if𝑖𝑗,0otherwise,for𝑖,𝑗=0,1,2,,𝑛.(2.1)

Definition 2.2. The 𝑛th order Wronskian matrix of 𝑓1(𝑡),𝑓2(𝑡),𝑓3(𝑡),, and 𝑓𝑚(𝑡) is an (𝑛+1) by 𝑚 matrix and is defined as 𝒲𝑛𝑓1(𝑡),𝑓2(𝑡),𝑓3(𝑡),,𝑓𝑚=𝑓(𝑡)1(𝑡)𝑓2(𝑡)𝑓3(𝑡)𝑓𝑚𝑓(𝑡)1(𝑡)𝑓2(𝑡)𝑓3(𝑡)𝑓𝑚(𝑓𝑡)1(𝑛)(𝑡)𝑓2(𝑛)(𝑡)𝑓3(𝑛)(𝑡)𝑓𝑚(𝑛)(𝑡).(2.2)

We study the Pascal functional and Wronskian matrices in a neighborhood of 𝑡=0. Hence, when we mention analytic, we mean analytic near 𝑡=0.

Note 1. Often, readers will encounter expressions such as 𝒫𝑛[𝑓(𝑥,𝑡)]𝑡=0 or 𝒲𝑛[𝑓(𝑥,𝑡)]𝑡=0. In this context, the variable 𝑡 is the working variable for the Pascal functional or the Wronkian matrix and the variable 𝑥 is merely a parameter.
In the following, we list some properties and relationships between the Pascal functional and Wronskian matrices that will be the main tool for our work.

Property 1. (a) 𝒫𝑛[] and 𝒲𝑛[] are linear, that is, for any constants 𝑎 and 𝑏, and any analytic functions 𝑓(𝑡) and 𝑔(𝑡), 𝒫𝑛[]𝑎𝑓(𝑡)+𝑏𝑔(𝑡)=𝑎𝒫𝑛[]𝑓(𝑡)+𝑏𝒫𝑛[],𝒲𝑔(𝑡)𝑛[]𝑎𝑓(𝑡)+𝑏𝑔(𝑡)=𝑎𝒲𝑛[𝑓](𝑡)+𝑏𝒲𝑛[𝑔].(𝑡)(2.3)
(b) For any analytic functions 𝑓(𝑡) and 𝑔(𝑡), 𝒫𝑛[]𝒫𝑓(𝑡)𝑛[]𝑔(𝑡)=𝒫𝑛[]𝒫𝑔(𝑡)𝑛[]𝑓(𝑡)=𝒫𝑛[]𝑓(𝑡)𝑔(𝑡).(2.4)
Furthermore, if 𝑓(𝑡)0, then (𝒫𝑛[𝑓(𝑡)])1=𝒫𝑛[𝑓1(𝑡)], where 𝑓1(𝑡) denotes the multiplicative inverse of 𝑓(𝑡).
(c) For any analytic functions 𝑓(𝑡) and 𝑔(𝑡), 𝒫𝑛[]𝒲𝑓(𝑡)𝑛[]𝑔(𝑡)=𝒫𝑛[]𝒲𝑔(𝑡)𝑛[]𝑓(𝑡)=𝒲𝑛[](𝑓𝑔)(𝑡).(2.5)
Furthermore, for any analytic functions 𝑔(𝑡), and 𝑓1(𝑡),𝑓2(𝑡),, and 𝑓𝑚(𝑡), 𝒫𝑛[𝑔]𝒲(𝑡)𝑛𝑓1(𝑡),𝑓2(𝑡),,𝑓𝑚(𝑡)=𝒲𝑛𝑔𝑓1(𝑡),𝑔𝑓2(𝑡),,𝑔𝑓𝑚(𝑡).(2.6)
(d) For any analytic functions 𝑔(𝑡) and 𝑓(𝑡) with 𝑓(0)=0 and 𝑓(0)0, 𝒲𝑛[]𝑔(𝑓(𝑡))𝑡=0=𝒲𝑛1,𝑓(𝑡),𝑓2(𝑡),𝑓3(𝑡),,𝑓𝑛(𝑡)𝑡=0Λ𝑛1𝒲𝑛[]𝑔(𝑡)𝑡=0,(2.7) where Λ𝑛=diag[0!,1!,2!,,𝑛!]. The notational convention Λ𝑛 will be used throughout this paper.

Proof. The proofs of Property 1(a), 1(b), and 1(c) can be found in [4].
For Property 1(d), let us express the functions 𝑓(𝑡) and 𝑔(𝑡) as series around 𝑡=0; 𝑓(𝑡)=𝑗=0𝑓𝑗𝑡𝑗 and 𝑔(𝑡)=𝑗=0𝑔𝑗𝑡𝑗. Since 𝑓(0)=0 and 𝑓(0)0, the leading term of 𝑓𝑘(𝑡) is 𝑡𝑘. Therefore, 𝑑𝑑𝑡(𝑘)||||𝑔(𝑓(𝑡))𝑡=0=𝑑𝑑𝑡(𝑘)𝑗=0𝑔𝑗𝑓𝑗|||||(𝑡)𝑡=0=𝑑𝑑𝑡𝑘(𝑘)𝑗=0𝑔𝑗𝑓𝑗|||||(𝑡)𝑡=0(2.8) because (𝑑/𝑑𝑡)(𝑘)𝑓𝑗(𝑡)|𝑡=0=0 for all 𝑗>𝑘. By (2.8) and noting 𝑔𝑘=𝑔(𝑘)(0)/𝑘!, we have 𝒲𝑛[]𝑔(𝑓(𝑡))𝑡=0=10000𝑓(0)000𝑓𝑓(0)2(0)00𝑓(𝑛)𝑓(0)2(𝑛)(0)(𝑓𝑛)(𝑛)𝑔(0)(0)(0)𝑔0!(1)(0)𝑔1!(2)(0)𝑔2!(𝑛)(0)𝑛!=𝒲𝑛1,𝑓(𝑡),𝑓2(𝑡),𝑓3(𝑡),,𝑓𝑛(𝑡)𝑡=0Λ𝑛1𝒲𝑛[𝑔](𝑡)𝑡=0.(2.9)
This completes the proof.

3. Sheffer Polynomial Sequence and Its Differential Equation

Let us first define a Sheffer polynomial sequence by a pair of generation functions (𝑔(𝑡),𝑓(𝑡)) as often done [5].

Definition 3.1. Let 𝑔(𝑡) be an invertible analytic function, that is, 𝑔(0)0, and 𝑓(𝑡) be analytic function with 𝑓(0)=0 and let 𝑓(0)0 that admits compositional inverse. Let 𝑓(𝑡) denote the compositional inverse of 𝑓(𝑡). Then, {𝑠𝑛(𝑥)} is the Sheffer polynomial sequence for (𝑔(𝑡),𝑓(𝑡)) if and only if 1𝑔𝑒𝑓(𝑡)𝑥𝑓(𝑡)=𝑘=0𝑠𝑘(𝑥)𝑡𝑘!𝑘.(3.1)

Note 2. Since (1/𝑔(𝑓(𝑡)))𝑒𝑥𝑓(𝑡) is analytic, by Taylor’s Theorem, 𝑠𝑘(𝑑𝑥)=𝑑𝑡(𝑘)1𝑔𝑒𝑓(𝑡)𝑥𝑓(𝑡)||||||𝑡=0.(3.2)

The family of Sheffer polynomial sequences contains two simpler subclasses of polynomial sequences, Appell and associated polynomial sequences [5]. An Appell polynomial sequence is a Sheffer polynomial sequence where 𝑓(𝑡)=𝑡. Hence, we say {𝑎𝑛(𝑥)} is the Appell polynomial sequence for 𝑔(𝑡) if and only if {𝑎𝑛(𝑥)} is the Sheffer polynomial sequence for (𝑔(𝑡),𝑡). Associated polynomial sequence is a Sheffer polynomial sequence where 𝑔(𝑡)=1. Hence, we say {𝑞𝑛(𝑥)} is the associated polynomial sequence for 𝑓(𝑡) if and only if {𝑞𝑛(𝑥)} is the Sheffer polynomial sequence for (1,𝑓(𝑡)).

We define the Sheffer polynomial sequence in vector form to utilize Wronskian matrices.

Definition 3.2. The Sheffer vector for (𝑔(𝑡),𝑓(𝑡)), denoted by 𝑆𝑛(𝑥), is defined as 𝑆𝑛𝑠(𝑥)=0(𝑥)𝑠1(𝑥)𝑠2(𝑥)𝑠𝑛(𝑥)𝑇,(3.3) where {𝑠𝑛(𝑥)} is the Sheffer polynomial sequence for (𝑔(𝑡),𝑓(𝑡)).
As noted in Note 2, the Sheffer vector can be expressed as 𝑆𝑛𝑠(𝑥)=0(𝑥)𝑠1(𝑥)𝑠2(𝑥)𝑠𝑛(𝑥)𝑇=𝒲𝑛1𝑔𝑒𝑓(𝑡)𝑥𝑓(𝑡)𝑡=0.(3.4)
In order to derive the differential equation for Sheffer polynomial sequence, we develop the following lemma.

Lemma 3.3. Let {𝑠𝑛(𝑥)} be the Sheffer polynomial sequence for (𝑔(𝑡),𝑓(𝑡)). Then, 𝒲𝑛𝑠0(𝑥),𝑠1(𝑥),,𝑠𝑛(𝑥)𝑇Λ𝑛1=𝒲𝑛1,𝑓(𝑡),𝑓2(𝑡),,𝑓𝑛(𝑡)𝑡=0Λ𝑛1𝒫𝑛1𝑔(𝑡)𝑡=0𝒫𝑛𝑒𝑥𝑡𝑡=0.(3.5)

Proof. Using (3.4) and Property 1(d), we have 𝑆𝑛(𝑥)=𝒲𝑛𝑒𝑥𝑓(𝑡)𝑔𝑓(𝑡)𝑡=0=𝒲𝑛1,𝑓(𝑡),𝑓2(𝑡),,𝑓𝑛(𝑡)𝑡=0Λ𝑛1𝒲𝑛𝑒𝑥𝑡𝑔(𝑡)𝑡=0.(3.6)
Using Property 1(c) and noting 𝒲𝑛[𝑒𝑥𝑡]𝑡=0=[1𝑥𝑥2𝑥𝑛]𝑇, we obtain 𝑆𝑛(𝑥)=𝒲𝑛1,𝑓(𝑡),𝑓2(𝑡),,𝑓𝑛(𝑡)𝑡=0Λ𝑛1𝒫𝑛1𝑔(𝑡)𝑡=01𝑥𝑥2𝑥𝑛𝑇.(3.7)
Taking the 𝑘th order derivative with respect to 𝑥 on both sides of (3.7) and dividing by 𝑘! yields 1𝑠𝑘!0(𝑘)(𝑥)𝑠1(𝑘)(𝑥)𝑠2(𝑘)(𝑥)𝑠𝑛(𝑘)(𝑥)𝑇=𝒲𝑛1,𝑓(𝑡),𝑓2(𝑡),,𝑓𝑛(𝑡)𝑡=0Λ𝑛1𝒫𝑛1𝑔(𝑡)𝑡=0×𝑘𝑥𝑘𝑥001𝑘+1𝑘+22𝑛𝑘𝑥𝑛𝑘𝑇.(3.8)
The left-hand of (3.8) is the 𝑘th column of 𝒲𝑛[𝑠0(𝑥),𝑠1(𝑥),,𝑠𝑛(𝑥)]𝑇Λ𝑛1, and the right-hand of (3.8) is the 𝑘th column of 𝑊𝑛[1,𝑓(𝑡),𝑓2(𝑡),,𝑓𝑛(𝑡)]𝑡=0Λ𝑛1𝒫𝑛[1/𝑔(𝑡)]𝑡=0𝒫𝑛[𝑒𝑥𝑡]𝑡=0.

After the introduction of definitions and the lemma, we are ready to develop differential equations for Sheffer polynomials.

Theorem 3.4. Let {𝑠𝑛(𝑥)} be the Sheffer polynomial sequence for (𝑔(𝑡),𝑓(𝑡)). Then, it satisfies the following differential equation: 𝑛𝑘=1𝛽𝑘𝑥+𝛼𝑘𝑠𝑛(𝑘)(𝑥)𝑘!𝑛𝑠𝑛(𝑥)=0,(3.9) where 𝛽𝑘=(𝑓(𝑡)/𝑓(𝑡))(𝑘)|𝑡=0 and 𝛼𝑘=(𝑔(𝑡)𝑓(𝑡)/𝑔(𝑡)𝑓(𝑡))(𝑘)|𝑡=0.

Proof. Let us consider 𝒲𝑛[𝑡(𝑑/𝑑𝑡)(𝑒𝑥𝑓(𝑡)/𝑔(𝑓(𝑡)))]𝑡=0. On one hand, by Property 1(c), we have 𝒲𝑛𝑡𝑑𝑒𝑑𝑡𝑥𝑓(𝑡)𝑔𝑓(𝑡)𝑡=0=𝒫𝑛[𝑡]𝑡=0𝒲𝑛𝑑𝑒𝑑𝑡𝑥𝑓(𝑡)𝑔𝑓(𝑡)𝑡=0=𝑠00000001000000020000000300000000𝑛10000000𝑛01𝑠(𝑥)2𝑠(𝑥)3𝑠(𝑥)𝑛𝑠(𝑥)𝑛+1.(𝑥)(3.10)
On the other hand, by Properties 1(c) and 1(d) and Lemma 3.3, we have𝒲𝑛𝑡𝑑𝑒𝑑𝑡𝑥𝑓(𝑡)𝑔𝑓(𝑡)𝑡=0=𝒲𝑛𝑔𝑥𝑓(𝑡)𝑔𝑡𝑓(𝑡)𝑓𝑒𝑓(𝑡)𝑥𝑓(𝑡)𝑔𝑓(𝑡)𝑡=0=𝒲𝑛1,𝑓(𝑡),,𝑓𝑛(𝑡)𝑡=0Λ𝑛1𝒲𝑛𝑔𝑥(𝑡)𝑔(𝑡)𝑓(𝑡)𝑓(𝑒𝑡)𝑥𝑡𝑔(𝑡)𝑡=0=𝒲𝑛1,𝑓(𝑡),,𝑓𝑛(𝑡)𝑡=0Λ𝑛1𝒫𝑛1𝑔(𝑡)𝑡=0𝒫𝑛𝑒𝑥𝑡𝑡=0×𝒲𝑛𝑥𝑓(𝑡)𝑓𝑔(𝑡)(𝑡)𝑓(𝑡)𝑔(𝑡)𝑓(𝑡)𝑡=0=𝒲𝑛𝑠0(𝑥),𝑠1(𝑥),,𝑠𝑛(𝑥)𝑇Λ𝑛1𝒲𝑛𝑥𝑓(𝑡)𝑓𝑔(𝑡)(𝑡)𝑓(𝑡)𝑔(𝑡)𝑓(𝑡)𝑡=0.(3.11)
Thus, 𝒲𝑛𝑡𝑑𝑒𝑑𝑡𝑥𝑓(𝑡)𝑔𝑓(𝑡)𝑡=0=𝑠0𝑠(𝑥)0001𝑠(𝑥)1(𝑥)𝑠1!002𝑠(𝑥)2(𝑥)𝑠1!2(𝑥)𝑠2!0𝑛𝑠(𝑥)𝑛(𝑥)𝑠1!𝑛(𝑥)𝑠2!𝑛(𝑛)(𝑥)𝛽𝑛!0𝑥+𝛼0𝛽1𝑥+𝛼1𝛽2𝑥+𝛼2𝛽𝑛𝑥+𝛼𝑛.(3.12)
Equating the last rows of (3.10) and (3.12), we get 𝑛𝑠𝑛(𝑥)=𝑛𝑘=0𝑠𝑛(𝑘)(𝑥)𝛽𝑘!𝑘𝑥+𝛼𝑘.(3.13)
Since 𝑓(0)=0, 𝛼0=𝛽0=0, a rearrangement of (3.13) produces the desired result.

The following corollaries are immediate consequences of Theorem 3.4. When 𝑔(𝑡)=1, 𝛼𝑘=0 for all 𝑘>0, and we have a differential equation for associated polynomials.

Corollary 3.5. Let {𝑞𝑛(𝑥)} be the associated polynomial sequence for 𝑓(𝑡). Then, it satisfies the following differential equation: 𝑛𝑘=1𝛽𝑘𝑥𝑞𝑛(𝑘)(𝑥)𝑘!𝑛𝑞𝑛(𝑥)=0,(3.14) where 𝛽𝑘=(𝑓(𝑡)/𝑓(𝑡))(𝑘)|𝑡=0.

Setting 𝑓(𝑡)=𝑡 in Theorem 3.4, we get a differential equation for Appell polynomials.

Corollary 3.6. Let {𝑎𝑛(𝑥)} be the Appell polynomial sequence for 𝑔(𝑡). Then, it satisfies the following differential equation: 𝑛𝑘=2𝛼𝑘1𝑎(𝑘1)!𝑛(𝑘)(𝑥)+𝑥+𝛼0𝑎𝑛(𝑥)𝑛𝑎𝑛(𝑥)=0,(3.15) where 𝛼𝑘=(𝑔(𝑡)/𝑔(𝑡))(𝑘)|𝑡=0.

Proof. Since 𝑓(𝑡)=𝑡, 𝛽1=1 and 𝛽𝑘=0 for all 𝑘1. Furthermore, 𝛼𝑘=𝑡𝑔(𝑡)𝑔(𝑡)(𝑘)||||𝑡=0=𝑘𝑔(𝑡)𝑔(𝑡)(𝑘1)||||𝑡=0=𝑘𝛼𝑘1.(3.16)
Hence, the above differential equation follows from Theorem 3.4.

Remark 3.7. The above differential equation for Appell is equivalent to the one in Theorem  2.1 in [2].
Let us apply Theorem 3.4 to derive differential equations of some well-known Sheffer polynomials. For the sake of brevity of context, we leave the detailed calculations of examples in the paper to interested readers.

Example 3.8. Let 𝐿𝑛{𝑎}(𝑥) denote the Laguerre polynomial of order 𝑎, which is the Sheffer polynomial for (𝑔(𝑡),𝑓(𝑡))=((1𝑡)𝑎1,𝑡/(𝑡1)). 𝐿𝑛{𝑎}(𝑥)=𝑛𝑘=0𝑛+𝑎𝑛𝑘(𝑛!/𝑘!)(𝑥)𝑘 satisfies the following differential equation: 𝑥𝐿𝑛{𝑎}(𝑥)𝐿(𝑥𝑎1)𝑛{𝑎}(𝑥)+𝑛𝐿𝑛{𝑎}(𝑥)=0.(3.17)
In particular for 𝑎=0, 𝐿𝑛{0}(𝑥)=𝐿𝑛(𝑥), generally known as the Laguerre polynomial, satisfies 𝑥𝐿𝑛(𝑥)(𝑥1)𝐿𝑛(𝑥)+𝑛𝐿𝑛(𝑥)=0.(3.18)

Example 3.9. Let 𝑝𝑛(𝑥) denote the Poisson-Charlier polynomial of order 𝑎, which is the Sheffer polynomial for (𝑔(𝑡),𝑓(𝑡))=(𝑒𝑎(𝑒𝑡1),𝑎(𝑒𝑡1)). 𝑝𝑛(𝑥)=𝑛𝑘=0(𝑛𝑘)(1)𝑛𝑘𝑎𝑘(𝑥)𝑘, where (𝑥)𝑘=𝑥(𝑥1)(𝑥2)(𝑥𝑘+1) is lower factorial polynomial, satisfies the following differential equation: 𝑛𝑘=1(1)𝑘+1𝑝𝑥𝑎𝑛(𝑘)(𝑥)𝑘!𝑛𝑝𝑛(𝑥)=0.(3.19)

Example 3.10. The Actuarial polynomial, denoted by 𝐴𝑛(𝑥), is the Sheffer polynomial for (𝑔(𝑡),𝑓(𝑡))=((1𝑡)𝑏,ln(1𝑡)). 𝐴𝑛(𝑥)=𝑛𝑘=0𝑏𝑘(𝑛𝑗=𝑘𝑆(𝑛,𝑗)(𝑗)𝑘(𝑥)𝑗𝑘), where 𝑆(𝑛,𝑘) is the Stirling number of the second kind and (𝑗)𝑘 is the lower factorial polynomial. Then, 𝑛𝑘=2𝑥+𝑏𝑘(𝑘1)𝑘𝐴𝑛(𝑘)(𝑥)(𝑥𝑏)𝐴𝑛(𝑥)+𝑛𝐴𝑛(𝑥)=0,for𝑛2.(3.20)

Example 3.11. The Lower factorial polynomial is the associated polynomial sequence for 𝑓(𝑡)=𝑒𝑎𝑡1. Lower factorial polynomial 𝑤𝑛(𝑥)=(𝑥/𝑎)𝑛=(𝑥/𝑎)(𝑥/𝑎1)(𝑥/𝑎𝑛+1) satisfies the following differential equation: 𝑥𝑛𝑘=1(𝑎)𝑘1𝑤𝑘!𝑛(𝑘)(𝑥)𝑛𝑤𝑛(𝑥)=0.(3.21)

Example 3.12. The exponential polynomial is the associated polynomial sequence for 𝑓(𝑡)=ln(1+𝑡). Let 𝜙𝑛(𝑥) denote the exponential polynomial and 𝜙𝑛(𝑥)=𝑛𝑘=0𝑆(𝑛,𝑘)𝑥𝑘, where 𝑆(𝑛,𝑘) is the Stirling number of the second kind. Then, 𝑥𝑛𝑘=2(1)𝑘𝜙𝑛(𝑘)(𝑥)𝑘(𝑘1)+𝑥𝜙𝑛(𝑥)𝑛𝜙𝑛(𝑥)=0,for𝑛2.(3.22)

Example 3.13. The Hermite polynomial of order 𝑣 is the Appell polynomial sequence for 𝑔(𝑡)=𝑒𝑣𝑡2/2. Let us denote the Hermite polynomial of order 𝑣 as 𝐻𝑛{𝑣}(𝑥) and 𝐻𝑛{𝑣}(𝑥)=𝑛/2𝑘=0(𝑣/2)𝑘((𝑛)2𝑘/𝑘!)𝑥𝑛2𝑘. Then 𝑣𝐻𝑛{𝑣}(𝑥)𝐻𝑥𝑛{𝑣}(𝑥)+𝑛𝐻𝑛{𝑣}(𝑥)=0.(3.23)

4. Recurrence Relations for the Sheffer Polynomials

Finding recursive formulas is one of main interests on study of the Sheffer polynomial sequences. For instance, Lehmer in [6] developed six recursive relations for the Bernoulli polynomial sequence (one of Appell polynomial sequences). In this section, we derive three recursive formulas. The first formula expresses 𝑠𝑛+1(𝑥) in terms of 𝑠𝑛(𝑥) and its derivatives, and the second and third formulas express 𝑠𝑛+1(𝑥) in terms of 𝑠𝑘(𝑥) for 𝑘=0,1,,𝑛.

4.1. The First Recursive Formula for the Sheffer Polynomials

Theorem 4.1 (Recursive Formula I). Let {𝑠𝑛(𝑥)} denote the Sheffer polynomial sequence for (𝑔(𝑡),𝑓(𝑡)). Then, 𝑠0(𝑥)=1/𝑔(0) and 𝑠𝑛+1(𝑥)=𝑛𝑘=0𝛾𝑘𝑥+𝛿𝑘𝑠𝑛(𝑘)(𝑥)𝑘!,for𝑛0,(4.1) where 𝛾𝑘=(1/𝑓(𝑡))(𝑘)|𝑡=0 and 𝛿𝑘=(𝑔(𝑡)/𝑔(𝑡)𝑓(𝑡))(𝑘)|𝑡=0.

Proof. The proof is similar to the proof of Theorem 3.4. Let us consider 𝒲𝑛[(𝑑/𝑑𝑡)(𝑒𝑥𝑓(𝑡)/𝑔(𝑓(𝑡)))]𝑡=0. On one hand, 𝒲𝑛𝑑𝑒𝑑𝑡𝑥𝑓(𝑡)𝑔𝑓(𝑡)𝑡=0=𝑠1(𝑥)𝑠2(𝑥)𝑠3(𝑥)𝑠𝑛+1(𝑥)𝑇.(4.2) (17
On the other hand, by Properties 1(c) and 1(d) and Lemma 3.3, we have𝒲𝑛𝑑𝑒𝑑𝑡𝑥𝑓(𝑡)𝑔𝑓(𝑡)𝑡=0=𝒲𝑛𝑔𝑥𝑓(𝑡)𝑔1𝑓(𝑡)𝑓𝑒𝑓(𝑡)𝑥𝑓(𝑡)𝑔𝑓(𝑡)𝑡=0=𝒲𝑛1,𝑓(𝑡),,𝑓𝑛(𝑡)𝑡=0Λ𝑛1𝒫𝑛1𝑔(𝑡)𝑡=0𝒫𝑛𝑒𝑥𝑡𝑡=0×𝒲𝑛𝑥1𝑓𝑔(𝑡)(𝑡)𝑔(𝑡)𝑓(𝑡)𝑡=0=𝒲𝑛𝑠0(𝑥),𝑠1(𝑥),,𝑠𝑛(𝑥)𝑇Λ𝑛1𝒲𝑛𝑥1𝑓𝑔(𝑡)(𝑡)𝑔(𝑡)𝑓(𝑡)𝑡=0.(4.3)
Thus, 𝒲𝑛𝑑𝑒𝑑𝑡𝑥𝑓(𝑡)𝑔𝑓(𝑡)𝑡=0=𝑠0𝑠(𝑥)0001𝑠(𝑥)1(𝑥)𝑠1!002𝑠(𝑥)2(𝑥)𝑠1!2(𝑥)𝑠2!0𝑛𝑠(𝑥)𝑛(𝑥)𝑠1!𝑛(𝑥)𝑠2!𝑛(𝑛)(𝑥)𝛾𝑛!0𝑥+𝛿0𝛾1𝑥+𝛿1𝛾2𝑥+𝛿2𝛾𝑛𝑥+𝛿𝑛.(4.4)
Equating the last rows of (17) and (4.4), we get the desired result.

Example 4.2. For 𝐴𝑛(𝑥)=𝑛𝑘=0𝑏𝑘(𝑛𝑗=𝑘𝑆(𝑛,𝑗)(𝑗)𝑘(𝑥)𝑗𝑘), the Actuarial polynomial, 𝐴0(𝑥)=1,𝐴𝑛+1(𝑥)=(𝑥+𝑏)𝐴𝑛(𝑥)+𝑥𝐴𝑛(𝑥),for𝑛0.(4.5)

Example 4.3. Let 𝐿𝑛(𝑥) denote the Laguerre polynomial. Then, 𝐿0(𝑥)=1,𝐿𝑛+1(𝑥)=(𝑥+1)𝐿𝑛(𝑥)+(2𝑥1)𝐿𝑛(𝑥)𝑥𝐿𝑛(𝑥),for𝑛0.(4.6)

Example 4.4. Hermite polynomial is defined as the Sheffer polynomial sequence for (𝑔(𝑡),𝑓(𝑡))=(𝑒𝑡2/4,𝑡/2) Let us denote the polynomial as 𝐻𝑛(𝑥). Then, 𝐻0(𝑥)=1,𝐻𝑛+1(𝑥)=2𝑥𝐻𝑛(𝑥)𝐻𝑛(𝑥),for𝑛0.(4.7)

Corollary 4.5 (Recursive Formula I for associated polynomial sequences). Let {𝑞𝑛(𝑥)} denote the associated polynomial sequence for 𝑓(𝑡). Then, 𝑞0(𝑥)=1 and 𝑞𝑛+1(𝑥)=𝑥𝑛𝑘=0𝛾𝑘𝑞𝑘!𝑛(𝑘)(𝑥),for𝑛0,(4.8) where 𝛾𝑘=(1/𝑓(𝑡))(𝑘)|𝑡=0.

Proof. It follows from Theorem 4.1 since 𝑔(𝑡)=1 and hence 𝛿𝑘=0 for all 𝑘.

Example 4.6. For the exponential polynomial 𝜙𝑛(𝑥), 𝜙0(𝑥)=1,𝜙𝑛+1𝜙(𝑥)=𝑥𝑛(𝑥)+𝜙𝑛(𝑥),for𝑛0.(4.9)

Corollary 4.7 (Recursive Formula I for Appell polynomial sequences). Let {𝑎𝑛(𝑥)} denote the Appell polynomial sequence for 𝑔(𝑡). Then, 𝑎0(𝑥)=1/𝑔(0) and 𝑎𝑛+1(𝑥)=𝑥𝑎𝑛(𝑥)+𝑛𝑘=0𝛿𝑘𝑎𝑘!𝑛(𝑘)(𝑥),for𝑛0,(4.10) where 𝛿𝑘=(𝑔(𝑡)/𝑔(𝑡))(𝑘)|𝑡=0.

Proof. Since 𝑓(𝑡)=𝑡, 𝛿𝑘=(𝑔(𝑡)/𝑔(𝑡)𝑓(𝑡))(𝑘)|𝑡=0=(𝑔(𝑡)/𝑔(𝑡))(𝑘)|𝑡=0,𝛾0=1, and 𝛾𝑘=0 for all other 𝑘 in Theorem 4.1.

Example 4.8. For the Hermite polynomial of order 𝑣, denoted by 𝐻𝑛{𝑣}(𝑥), 𝐻0{𝑣}(𝑥)=1,𝐻{𝑣}𝑛+1(𝑥)=𝑥𝐻𝑛{𝑣}𝐻(𝑥)𝑣𝑛{𝑣}(𝑥),for𝑛0.(4.11)

Example 4.9. Let us consider the Stirling polynomial sequence 𝑆𝑛(𝑥), which is the Sheffer polynomial sequence for (𝑔(𝑡),𝑓(𝑡))=(𝑒𝑡,ln(𝑡/1𝑒𝑡)). To obtain the recursive formula for 𝑆𝑛(𝑥) by Theorem 4.1, we have to find 𝑓(𝑡), that is, solve the insolvable transcendental equation 𝑒𝑡=𝑦1𝑒𝑦(4.12) for 𝑦.

This example shows a type of Sheffer sequences for which Theorem 4.1 fails to produce a recursive formula. This motivate us to develop other recursive formulas, which represent 𝑠𝑛+1(𝑥) in term of its previous terms 𝑠𝑘(𝑥) and the derivatives of 𝑓(𝑡) and 𝑔(𝑡).

4.2. The Second Recursive Formula for the Sheffer Polynomials

Theorem 4.10 (Recursive Formula II). Let {𝑠𝑛(𝑥)} be the Sheffer polynomial sequence for (𝑔(𝑡),𝑓(𝑡)). Then, 𝑠0(𝑥)=1/𝑔(0) and 𝜖0𝑠𝑛+1(𝑥)=𝑥𝑠𝑛(𝑥)+𝑛𝑘=0𝑛𝑘𝜃𝑘𝑠𝑛𝑘(𝑥)𝑛𝑘=1𝑛𝑘𝜖𝑘𝑠𝑛+1𝑘(𝑥),for𝑛0,(4.13) where 𝜖𝑘=(𝑓(𝑓(𝑡)))(𝑘)|𝑡=0=(1/𝑓(𝑡))(𝑘)|𝑡=0𝑎𝑛𝑑𝜃𝑘=(𝑔(𝑓(𝑡))/𝑔(𝑓(𝑡)))(𝑘)|𝑡=0.

Proof. Let us consider 𝒲𝑛[𝑓(𝑓(𝑡))(𝑑/𝑑𝑡)(𝑒𝑥𝑓(𝑡)/𝑔(𝑓(𝑡)))]𝑡=0. On one hand, applying Property 1(c),𝒲𝑛𝑓𝑑𝑓(𝑡)𝑒𝑑𝑡𝑥𝑓(𝑡)𝑔𝑓(𝑡)𝑡=0=𝒫𝑛𝑑𝑒𝑑𝑡𝑥𝑓(𝑡)𝑔𝑓(𝑡)𝑡=0𝒲𝑛𝑓𝑓(𝑡)𝑡=0=𝑠1𝑠(𝑥)0002(𝑥)𝑠1𝑠(𝑥)00321𝑠(𝑥)2(𝑥)𝑠1𝑠(𝑥)0𝑛+1𝑛1𝑠(𝑥)𝑛𝑛2𝑠(𝑥)𝑛1(𝑥)𝑠1𝜖(𝑥)0𝜖1𝜖2𝜖𝑛.(4.14)
On the other hand, by Properties 1(a) and 1(c), 𝒲𝑛𝑓𝑑𝑓(𝑡)𝑒𝑑𝑡𝑥𝑓(𝑡)𝑔𝑓(𝑡)𝑡=0=𝒲𝑛𝑔𝑥𝑓(𝑡)𝑔𝑒𝑓(𝑡)𝑥𝑓(𝑡)𝑔𝑓(𝑡)𝑡=0=𝑥𝒲𝑛𝑒𝑥𝑓(𝑡)𝑔𝑓(𝑡)𝑡=0𝒫𝑛𝑒𝑥𝑓(𝑡)𝑔𝑓(𝑡)𝑡=0𝒲𝑛𝑔𝑓(𝑡)𝑔𝑓(𝑡)𝑡=0𝑠=𝑥0𝑠(𝑥)1𝑠(𝑥)2(𝑠𝑥)𝑛+𝑠(𝑥)0𝑠(𝑥)0001(𝑥)𝑠0𝑠(𝑥)00221𝑠(𝑥)1(𝑥)𝑠0𝑠(𝑥)0𝑛(𝑛1𝑠𝑥)𝑛1(𝑛2𝑠𝑥)𝑛2(𝑥)𝑠0(𝜃𝑥)0𝜃1𝜃2𝜃𝑛.(4.15)
Equating the last rows of (4.14) and (4.15) leads to 𝑛𝑘=0𝑛𝑘𝑠𝑛+1𝑘(𝑥)𝜖𝑘=𝑥𝑠𝑛(𝑥)+𝑛𝑘=0𝑛𝑘𝑠𝑛𝑘(𝑥)𝜃𝑘.(4.16)
A rearrangement of the above yields the desired result.

Example 4.11. Let 𝐿𝑛{𝑎}(𝑥) be the Laguerre polynomial of order 𝑎, 𝑝𝑛(𝑥) the Poisson-Charlier polynomial of order 𝑎, and 𝑀𝑛(𝑥) the Meixner polynomial of the first kind of order (𝑏,𝑐), which is the Sheffer polynomial sequence for (𝑔(𝑡),𝑓(𝑡))=(((1𝑐)/(1𝑐𝑒𝑡))𝑏,(1𝑒𝑡)/(𝑐1𝑒𝑡)). Then, 𝐿0{𝑎}(𝑥)=1,𝐿{𝑎}𝑛+1(𝑥)=(2𝑛+𝑎+1𝑥)𝐿𝑛{𝑎}𝑛(𝑥)2𝐿+𝑛𝑎{𝑎}𝑛1𝑝(𝑥),for𝑛0,0(𝑥)=1,𝑎𝑝𝑛+1(𝑥)=(𝑥𝑛𝑎)𝑝𝑛(𝑥)𝑛𝑝𝑛1𝑀(𝑥),for𝑛0,0(𝑥)=1,𝑐𝑀𝑛+1(𝑥)=((𝑐1)𝑥+𝑛(𝑐+1)+𝑏𝑐)𝑀𝑛(𝑥)(𝑛(𝑛1)+𝑛𝑏)𝑀𝑛1(𝑥),for𝑛0.(4.17)

Corollary 4.12 (Recursive Formula II for associated polynomial sequences). Let {𝑞𝑛(𝑥)} denote the associated polynomial sequence for 𝑓(𝑡). Then, 𝑞0(𝑥)=1 and 𝜖0𝑞𝑛+1(𝑥)=𝑥𝑞𝑛(𝑥)𝑛𝑘=1𝑛𝑘𝜖𝑘𝑞𝑛+1𝑘(𝑥),for𝑛0,(4.18) where 𝜖𝑘=(𝑓(𝑓(𝑡)))(𝑘)|𝑡=0.

Example 4.13. Let 𝜙𝑛(𝑥) denote the exponential polynomial. Then, 𝜙0(𝑥)=1,𝜙𝑛+1(𝑥)=𝑥𝜙𝑛(𝑥)+𝑛1𝑘=0𝑛𝑘+1(1)𝑘𝜙𝑛𝑘(𝑥),for𝑛0.(4.19)

Corollary 4.14 (Recursive Formula II for Appell polynomial sequences). Let {𝑎𝑛(𝑥)} denote the Appell polynomial sequence for 𝑔(𝑡). Then, 𝑎0(𝑥)=1/𝑔(0) and 𝑎𝑛+1(𝑥)=𝑥𝑎𝑛(𝑥)+𝑛𝑘=0𝑛𝑘𝜃𝑘𝑎𝑛𝑘(𝑥),for𝑛0,(4.20) where 𝜃𝑘=(𝑔(𝑡)/𝑔(𝑡))(𝑘)|𝑡=0.

Remark 4.15. Theorem  2.2 and Theorem  2.4 in [2] are special cases of Corollary 4.14.

Example 4.16. Let 𝐻𝑛{𝑣}(𝑥) denote the Hermite polynomial of order 𝑣. Then, 𝐻0{𝑣}(𝑥)=1,𝐻{𝑣}𝑛+1(𝑥)=𝑥𝐻𝑛{𝑣}(𝑥)𝑛𝑣𝐻{𝑣}𝑛1(𝑥),for𝑛0.(4.21)

Here, we would like to revisit Example 4.9. In order to obtain the recursive formula for the Stirling polynomial sequence {𝑆𝑛(𝑥)} by Theorem 4.10, we need to compute 𝜖𝑘=𝑓𝑓(𝑡)(𝑘)||||𝑡=0=1𝑓(𝑡)(𝑘)|||||𝑡=0=𝑡𝑒𝑡1𝑒𝑡1𝑡(𝑘)|||||𝑡=0.(4.22)

Equation (4.22) seems difficult to evaluate and does not lead to any nice result. This is a motivation for us to develop yet another formula in Theorem 4.17.

4.3. The Third Recursive Formula for the Sheffer Polynomials

Theorem 4.17 (Recursive Formula III). Let {𝑠𝑛(𝑥)} denote the Sheffer polynomial sequence for (𝑔(𝑡),𝑓(𝑡)). Then, 𝑠0(𝑥)=1/𝑔(0) and 𝑠𝑛+1(𝑥)=𝑛𝑘=0𝑛𝑘𝑥𝛼𝑘+𝛽𝑘𝑠𝑛𝑘(𝑥),for𝑛0,(4.23) where 𝛼𝑘=(1/𝑓(𝑓(𝑡)))(𝑘)|𝑡=0=(𝑓(𝑡))(𝑘+1)|𝑡=0𝑎𝑛𝑑𝛽𝑘=(𝑔(𝑓(𝑡))/𝑔(𝑓(𝑡))𝑓(𝑓(𝑡)))(𝑘)|𝑡=0.

Proof. We have 𝑠1(𝑥)𝑠2(𝑥)𝑠3(𝑥)𝑠𝑛+1(𝑥)𝑇=𝒲𝑛𝑑𝑒𝑑𝑡𝑥𝑓(𝑡)𝑔𝑓(𝑡)𝑡=0.(4.24)
Also, 𝒲𝑛𝑑𝑒𝑑𝑡𝑥𝑓(𝑡)𝑔𝑓(𝑡)𝑡=0=𝒲𝑛𝑔𝑥𝑓(𝑡)𝑔1𝑓(𝑡)𝑓𝑒𝑓(𝑡)𝑥𝑓(𝑡)𝑔𝑓(𝑡)𝑡=0=𝒫𝑛𝑔𝑥𝑓(𝑡)𝑔1𝑓(𝑡)𝑓(𝑓(𝑡))𝑡=0𝒲𝑛𝑒𝑥𝑓(𝑡)𝑔𝑓(𝑡)𝑡=0=𝑥𝛼0+𝛽0000𝑥𝛼1+𝛽1𝑥𝛼0+𝛽000𝑥𝛼2+𝛽22𝑥𝛼1+𝛽1𝑥𝛼0+𝛽00𝑥𝛼𝑛+𝛽𝑛𝑛1𝑥𝛼𝑛1+𝛽𝑛1𝑛2𝑥𝛼𝑛2+𝛽𝑛2𝑥𝛼0+𝛽0𝑠0𝑠(𝑥)1(𝑠𝑥)2𝑠(𝑥)𝑛.(𝑥)(4.25)
Equating the last rows of (4.24) and (4.25), we get the desired result.

Example 4.18. Let 𝐴𝑛(𝑥) be the Actuarial polynomial. Then, 𝐴0(𝑥)=1,𝐴𝑛+1(𝑥)=𝑥𝑛𝑘=0𝑛𝑘𝐴𝑘(𝑥)+𝑏𝐴𝑛(𝑥),for𝑛0.(4.26)

Corollary 4.19 (Recursive Formula III for associated polynomial sequences). Let {𝑞𝑛(𝑥)} denote the associated polynomial sequence for 𝑓(𝑡). Then, 𝑞0(𝑥)=1 and 𝑞𝑛+1(𝑥)=𝑥𝑛𝑘=0𝑛𝑘𝛼𝑘𝑞𝑛𝑘(𝑥),for𝑛0,(4.27) where 𝛼𝑘=(1/𝑓(𝑓(𝑡)))(𝑘)|𝑡=0=(𝑓(𝑡))(𝑘+1)|𝑡=0.

Example 4.20. Let 𝜙𝑛(𝑥) denote the exponential polynomial. By Corollary 4.19, we have the following well-known result 𝜙0(𝑥)=1,𝜙𝑛+1(𝑥)=𝑥𝑛𝑘=0𝑛𝑘𝜙𝑘(𝑥),for𝑛0.(4.28)

Finally, let us finish up Example 4.9 and conclude this paper.

To obtain the recursive formula for the Stirling polynomial sequence {𝑆𝑛(𝑥)}by Theorem 4.17, we need to compute𝛼𝑘=1𝑓𝑓(𝑡)(𝑘)||||||𝑡=0=𝑓(𝑡)(𝑘+1)||||𝑡=0=1𝑡1𝑒𝑡1(𝑘)||||𝑡=0𝛽𝑘=𝑔𝑓(𝑡)𝑔𝑓𝑓(𝑡)𝑓(𝑡)(𝑘)|||||||𝑡=0=1𝑡1𝑒𝑡1(𝑘)||||𝑡=0.(4.29) Noting 𝑡/(𝑒𝑡1) is the exponential generating function for Bernoulli number sequence {𝑛} in [7], we can easily compute (4.29) to get𝛼𝑘=𝛽𝑘=1𝑡1𝑒𝑡1(𝑘)||||𝑡=01𝑡𝑡1𝑒𝑡1(𝑘)||||𝑡=0=𝑗=0𝑗+1𝑡(𝑗+1)!𝑗(𝑘)|||||𝑡=0=𝑘+1.𝑘+1(4.30)

Therefore, the recursive formula for the Stirling polynomial 𝑆𝑛(𝑥) is 𝑆0(𝑥)=1 and𝑆𝑛+1(𝑥)=𝑛𝑘=0𝑛𝑘(𝑥+1)𝑘+1𝑆𝑘+1𝑛𝑘((𝑥)=𝑥+1)𝑛+1𝑛𝑘=0𝑛+1𝑘+1𝑘+1𝑆𝑛𝑘(𝑥).(4.31)

References

  1. A. di Bucchianico and D. Loeb, “A selected survey of umbral calculus,” Electronic Journal of Combinatorics, vol. 2, pp. 1–34, 2000. View at Scopus
  2. M. X. He and P. E. Ricci, “Differential equation of Appell polynomials via the factorization method,” Journal of Computational and Applied Mathematics, vol. 139, no. 2, pp. 231–237, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  3. Y. Yang and C. Micek, “Generalized Pascal functional matrix and its applications,” Linear Algebra and Its Applications, vol. 423, no. 2-3, pp. 230–245, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  4. Y. Yang and H. Youn, “Appell polynomial sequences: a linear algebra approach,” JP Journal of Algebra, Number Theory and Applications, vol. 13, no. 1, pp. 65–98, 2009.
  5. S. Roman, The Umbral Calculus, Academic Press, Orlando, Fla, USA, 1984.
  6. D. H. Lehmer, “A new approach to Bernoulli polynomials,” The American Mathematical Monthly, vol. 95, no. 10, pp. 905–911, 1988. View at Publisher · View at Google Scholar · View at MathSciNet
  7. K. Rosen, Handbook of Discrete and Combinatorial Mathematics, CRC Press LLC, Boca Raton, Fla, USA, 2000.