Binomial transform and the backward difference

Abstract

We prove an important property of the binomial transform: it converts multiplication by the discrete variable into a certain difference operator. We also consider the case of dividing by the discrete variable. The properties presented here are used to compute various binomial transform formulas involving harmonic numbers, skew-harmonic numbers, Fibonacci numbers, and Stirling numbers of the second kind. Several new identities are proved and some known results are given new short proofs.

The paper is a slight modification of the journal article in Advances and Applications in Discrete Mathematics, 13 (1) (2014), 43-63