Binomial transform and the backward difference

TL;DR

This paper explores the properties of the binomial transform, demonstrating its ability to convert multiplication into difference operations and applying it to derive identities involving special number sequences.

Contribution

It introduces new properties of the binomial transform related to discrete variable operations and provides new proofs for identities involving harmonic, Fibonacci, and Stirling numbers.

Findings

Proves binomial transform converts multiplication into difference operations

Derives new identities involving harmonic and Fibonacci numbers

Provides short proofs for known identities

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.