Notes on the Binomial Transform
by Khristo N. Boyadzhiev
World Scientific, 2018
Transforms of different kinds, especially integral transforms, play an important role in various areas of mathematics. Discrete transforms, on the other hand, transform a given sequence, which may or may not be an integer sequence, to another sequence. One of the most important examples of such a transform is the binomial transform, the topic of this book. The binomial transform can be seen as a higher-order extension of the basic (forward) difference operator. It is also related to the Euler transform and to exponential generating functions. The binomial transform has a beautiful inversion formula, which gives rise to a great deal of sctructure, and to connections with various important objects and sequences in combinatorics. Much of this is collected in a concise way in this fairly slim volume.
The contents of this book are best described by quoting from the Preface: “The binomial transform leads to various combinatorial and analytical identities involving binomial coefficients. In particular, we present here new binomial identities for Bernoulli, Fibonacci, and harmonic numbers. Many interesting identities can be written as binomial transforms and vice versa.
“The volume consists of two parts. In the first part, we present the theory of the binomial transform for sequences with a sufficient prerequisite of classical numbers and polynomials. The first part provides theorems and tools which help to compute binomial transforms of different sequences and also to generate new binomial identities from the old. These theoretical tools (formulas and theorems) can also be used for summation of series and various numerical computations.
“In the second part, we have compiled a list of binomial transform formulas for easy reference. In the Appendix, we present the definition of the Stirling sequence transform and a short table of transformation formulas.”
This book will be useful reference to researchers interested in enumerative combinatorics, special number and polynomial sequences, and combinatorial number theory.