Notes on the Binomial Transform

The binomial transform is a discrete transformation of one sequence into another with many interesting applications in combinatorics and analysis. This volume is helpful to researchers interested in enumerative combinatorics, special numbers, and classical analysis. A valuable reference, it can also be used as lecture notes for a course in binomial identities, binomial transforms and Euler series transformations. 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.