The Umbral Calculus

PREFACE

This monograph is intended to be an elementary introduction to the modern umbral calculus. Since we have in mind the largest possible audience, the only prerequisite is an acquaintance with the basic notions of algebra, and perhaps a dose of applied mathematics (such as differential equations) to help put the theory in some mathematical perspective.

The title of this work really should have been The Modern Classical Umbral Calculus. Within the past few years many, indeed infinitely many, distinct umbral calculi have begun to be studied. Actually, the existence of distinct umbral calculi was recognized in a vague way as early as the 1930s but seems to have remained largely ignored until the past decade.

In any case, we shall occupy the vast majority of our time in studying one particular umbral calculus—the one that dates back to the 1850s and that has received the attention (both good and bad) of mathematicians up to the present time. For this, we use the term classical umbral calculus. Only in the last chapter do we glimpse the newer, much less well established, nonclassical umbral calculi.

The classical umbral calculus, as it was from 1850 to about 1970, consisted primarily of a symbolic technique for the manipulation of sequences, whose mathematical rigor left much to be desired. To drive this point home one need only look at Eric Temple Bell’s unsuccessful attempt (in 1940) to convince the mathematical community to accept the umbral calculus as a legitimate mathematical tool. (Even now some are still trying to achieve Eric Temple Bell’s original goal.) This old-style umbral calculus was, however, useful in deriving certain mathematical results; but unfortunately these results had to be verified by a different, more rigorous method.

In the 1970s Gian-Carlo Rota, a mathematician with a superlative talent for handling just this sort of situation, began to construct a completely rigorous foundation for the theory — one that was based on the relatively modern ideas of a linear functional, a linear operator, and an adjoint. In 1977, the author was fortunate enough to join in on this development.

It is this modern classical umbral calculus that is the subject of the present monograph.

Perusal of the table of contents will give the reader an idea of the organization of the book; but let us make a few remarks in this regard. A choice had to be made between the present organization of Chapters 2–4 and the alternative of integrating these chapters by applying each new aspect of the theory to a running list of examples. We feel that the alternative approach has a tendency to minimize the effect of the theory, making it difficult to see just what the umbral calculus can do in a specific instance. On the other hand, we recognize that it can be difficult to remain motivated in the face of a large dose of theory, untempered by any examples. For this reason, we have included at the end of Chapter 2 a very brief discussion of some of the more accessible examples. Chapter 4 contains a more complete discussion of these and other examples. Let us emphasize, however, that we do not intend this book to be a treatise on any particular polynomial sequence, nor do we make any claims concerning the originality of the formulas contained herein. While Chapter 2 contains the définition and general properties of the principal object of study—the Sheffer sequence— it is Chapter 3 that really goes to the heart of the modern umbral method. In Chapter 6 we touch on some of the nonclassical umbral calculi, but only enough to whet the appetite for, it is hoped, a sequel to this volume.

Before we begin, we should like to express our gratitude to Professor Gian-Carlo Rota. His help and encouragement have proved invaluable over the years.

CHAPTER 1

INTRODUCTION

1. A DEFINITION OF THE CLASSICAL UMBRAL CALCULUS

Sequences of polynomials play a fundamental role in applied mathematics. Such sequences can be described in various ways, for example,

(1) by orthogonality conditions:

where w(x) is a weight function and δnm = 0 or 1 according as n ≠ m or n = m;

(2) as solutions to differential equations: for instance, the Hermite polynomials Hn(x) satisfy the second-order linear differential equation

(3) by generating functions: for instance, the Bernoulli polynomials are characterized by

(4) by recurrence relations: as an example, the exponential polynomials ϕn(x) satisfy

(5) by operational formulas: for example, the Laguerre polynomials satisfy

(some put n! on the left).

One of the simplest classes of polynomial sequences, yet still large enough to include many important instances, is the class of Sheffer sequences (also known, in a slightly different form, as sequences of Sheffer A-type zero or poweroids). This class may be defined in many ways, most commonly by a generating function and, as Sheffer himself did, by a differential recurrence relation. Although we shall not adopt either of these means of définition, let us point out now that a sequence sn(x) is a Sheffer sequence if and only if its generating function has the form

where

and

The Sheffer class contains such important sequences as those formed by

(1) the Hermite polynomials, which play an important role in applied mathematics and physics (such as Brownian motion and the Schrödinger wave equation);

(2) the Laguerre polynomials, which also play a key role in applied mathematics and physics (they are involved in solutions to the wave equation of the hydrogen atom);

(3) the Bernoulli polynomials, which find applications, for example, in number theory (evaluation of the Hurwitz zeta function, a generalization of the famous Riemann zeta function);

(4) the Abel polynomials, which have a connection with geometric probability (the random placement of nonoverlapping arcs on a circle);

(5) the central factorial polynomials, which play a role in the interpolation of functions.

Now to the point at hand. The modern classical umbral calculus can be described as a systematic study of the class of Sheffer sequences, made by employing the simplest techniques of modern algebra.

More explicitly, if P is the algebra of polynomials in a single variable, the set P* of all linear functional on P is usually thought of as a vector space (under pointwise operations). However, it is well known that a linear functional on P can be identified with a formal power series. In fact, there is a one-to-one correspondence between linear functionals on P and formal power series in a single variable. For example, we may associate to each linear functional L the power series But the set of formal power series is usually given the structure of an algebra (under formal addition and multiplication). This algebra structure, the additive part of which agrees with the vector space structure on P*, can be transferred to P*. The algebra so obtained is called the umbral algebra, and the umbral calculus is the study of this algebra.

As a first step in this direction, since P* now has the structure of an algebra, we may consider, for two linear functional L and M, the geometric sequence M, ML, ML², ML³,…. Then under mild conditions on L and M, the equations

for n, k ≥ 0 uniquely determine a sequence sn(x) of polynomials which turns out to be of Sheffer type, and, conversely, for any sequence sn(x) of Sheffer type there are linear functionals L and M for which the above equations hold. Thus we may characterize the class of Sheffer sequences by means of the umbral algebra. The resulting interplay between the umbral algebra and the algebra of polynomials allows for the natural development of some powerful adjointness properties wherein lies the real strength of the theory.

The umbral calculus is, to be sure, formal mathematics. By this we mean that limiting processes, such as the convergence of infinite series, play no role. Formal mathematics, much of which comes under the headings of combinatorics, the calculus of finite differences, the theory of special functions, and formal solutions to differential equations, is, in the opinion of some, staging a comeback after many years of neglect. It is our hope that the present work will aid in this comeback.

2. PRELIMINARIES

Since formal power series play a predominant role in the umbral calculus, we should set down some basic facts concerning their use. The simple proofs either can be supplied by the reader or can be gleaned from other sources, such as the paper of Niven [1].

Let C be a field of characteristic zero. Let be the set of all formal power series in the variable t over C. Thus an element of has the form

for ak in