Umbral calculus originated as a method for discovering and proving combinatorial identities, but is developed in a more general form in this book. The present volume is a 2005 Dover unaltered reprint of the 1984 Academic Press edition.

Umbral calculus was developed in the 1800s and is attributed to various combinations of John Blissard, Édouard Lucas, and James Joseph Sylvester. The early days of umbral calculus were like the early days of the infinitesimal calculus: in the hands of skilled practitioners it produced correct results, but no one was sure why it worked. E. T. Bell was fascinated by the umbral calculus and attempted a revival of it in the 1930s and 1940s, but still no one understood it well. Gian-Carlo Rota became interested in it in the 1960s and 1970s and put a more general form of umbral calculus on a rigorous basis. Steven Roman, the author of the present work, was one of Rota’s collaborators, and this book explains this formal theory.

The term umbral was introduced by Sylvester from umbra, Latin for shadow. A familiar example of the umbral notation is the mnemonic for the Bernoulli numbers, \( (B+1)^n - B^n = 0\),

in which, after expanding by the binomial theorem, we replace \(B^k\) with \(B_k\) to get \(\sum_{k=0}^n \binom{n}{k} B_k - B_n = 0\), a recursive formula for the Bernoulli numbers. The symbol \(B\) is called the umbra, and in a nutshell the umbral calculus consisted of this special treatment of the umbra, where powers are replaced by subscripts. This example is just a mnemonic, but the umbral calculus actually can perform manipulations and create new results. The present book conceptualizes in a different way that does not involve powers or subscripts.

The classical way to justify this Bernoulli formula is through exponential generating functions. The umbral calculus presented in this book uses the same idea but formalizes it differently, in terms of linear functionals on the linear space of polynomials. Instead of the exponential generating function \(\sum a_n x^n / n!,\) we use a functional \(L\) defined by \(L(x^n) = a_n\). Conversely, any functional on the polynomials has a corresponding exponential generating function. As usual, the functionals form a linear space (the dual space), but we turn this into an algebra by defining a multiplication corresponding to the Cauchy product of coefficients that results from multiplying two power series, so the product of two functionals matches the product of the corresponding series. The book then develops several standard operators (functionals) and their properties.

This method is applied to sequences of polynomials (such as the Bernoulli polynomials), where each polynomial has a recursive relationship to the next one in the sequence, such as being a constant times its derivative. We then need a corresponding sequence of linear functionals of a particular form, so that each one takes the corresponding polynomial to \(n!\), and takes all other polynomials to zero. This sounds very specialized, but in fact it can be applied to many polynomial sequences that are useful in applications. The book develops a collection of general theorems about relations among the polynomials that apply if the set-up is satisfied.

The development here is strictly algebraic; it uses the notation of functional analysis but not any analytic properties. One impressive achievement of the book, though, is a development of the Lagrange reversion formula without using any complex analysis. The functional approach does not prove anything that could not be proved by generating functions, but it is easier to generalize and the operator approach may be easier to think about, especially since many of the results involve factoring operators.

Nearly half the book is in Chapter 4, that applies theses general theorems for particular sequences of polynomials, such as the Hermite, Laguerre, and Bernoulli polynomials. The book does not cover anything from combinatorics per se. This is primarily a special functions book. The methods shown here can be used to prove combinatorial identities, and Rota did this in a series of papers, but this aspect is barely mentioned in the present book.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.