Customer Reviews

The umbral calculus, Volume 111 (Pure and Applied Mathematics)

5 star:		(3)
4 star:		(1)
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I'd give this about four-and-a-half stars, but the software allows only integers. In the preface, the author, Steven Roman, correctly identifies this book's deficiency quite explicitly: It presents the theory before explaining the motivation. Later, when he gets into application to concrete examples, Roman also seems to view such application as the only source of motivation. The book is primarily about Sheffer sequences. A Sheffer sequence is a certain kind of polynomial sequence. A polynomial sequence is understood to be a sequence indexed by the nonnegative integers, in which the index equals the degree. Many of the well-known special polynomial sequences are Sheffer sequences, among them the Hermite polynomials, the Laguerre polynomials, the Abel polynomials, the Touchard (or "exponential") polynomials, etc. The concept of Sheffer sequence can be characterized by saying that the linear transformation on polynomials that maps the nth polynomial in the sequence to n times the (n-1)th polynomial, is shift-equivariant. It can also be characterized in a colorful way by the relationship of each Sheffer sequence to a polynomial sequence of binomial type. But neither of those is the definition that Roman gives. His definition is not motivated, until and unless you eventually figure out why it's the right thing by digesting the theory, as you eventually will if you keep at it. The solution to the motivation problem is to read "Finite Operator Calculus" by Rota, Kahaner, and Odlyzko. That paper, published in 1973, appeared in a book of the same title in 1975. That remarkable paper ends with a huge list of research problems, many of which, if I'm not mistaken, have not yet been addressed. Roman's book works through the theory, and its application to many examples, with great thoroughness.

"The umbral calculus" is traditionally the name of what is also called "Blissard's symbolic method", which is a notational device for proving identities involving indexed sequences of numbers by pretending the indices are exponents. The flavor of the technique can be seen by looking at John Riordan's book _Combinatorial_Identities_, which uses it extensively. In what sense that weird notational technique is the same thing as the study of Sheffer sequences, which is the subject of Roman's book, is something the reader should think about.

Another viewpoint on this subject than the one in the paper of Rota, Kahaner, and Odlyzko is in a paper titled _The_Umbral_Calculus_, by Steven Roman and Gian-Carlo Rota, published in Advances in Mathematics, volume 31, pages 95-188, in 1978. That paper makes the connection between the theory of Sheffer sequences and the "symbolic method" of Blissard much clearer than does the one by R., K., & O., and can also serve as motivation if you need that before tackling this book.

I'm writing this second review in order to mention that other source, but, since I earlier said I'd give this four-and-a-half stars, I'm taking this opportunity to raise the average to four-and-a-half by giving it five stars this time.

Have you come across the umbral calculus? The umbra (Latin: "shadow") is the darkest part of a shadow. But in mathematics? Perhaps a generating function is the shadow of its distribution? Or the term umbral calculus refers to surprising similarities between otherwise unrelated polynomial equations, and certain shadowy techniques used to prove them.

It is used in the study of orthogonal polynomials, in differential equations, in computations generally, and in the study of probability distributions. In fact it is an old (classical!) theory, but it undergoes periodic revivals.

An especially notable revival was pioneered by Gian-Carlo Rota in the 1970ties. He put the subject on a firm (rigorous!) foundation; with axioms, operators, diagrams, and duality arrows.

Previously it had been more like a bag of tricks involving formal power series and ad hoc computations with generating functions.

Motivated by Rota, Steven Roman came out with a different book: offering a wider view, a striking elegance, and including numerous new and intriguing applications. Roman's book was now reprinted in 2005 by Dover in this attractive little volume. Review by Palle Jorgensen, September 2007.

This book is a reference text on polynomials that are useful in quantum mechanics. Orthogonal polynomials are the backbone of Hilbert spaces and solutions to problems like the wave mechanics of hydrogen.

The only problem I have with the book is that he has only 5 categories and leaves out the matrix generated polynomials that are related to

to Heisenberg's matrix mechanics which is equivalent to Schrödinger's wave mechanics. I wish that I could already say I understood things like Vandermonde's convolution formula,

but that is why I bought the book

and why it delivers the goods.