Binomial Transforms and the Binomial Convolution of Sequences

TL;DR

This paper explores the relationships between binomial transforms and convolutions of sequences, deriving new identities involving special number sequences and methods to generate new transform pairs.

Contribution

It introduces simple relations connecting binomial convolutions with binomial transforms, enabling the derivation of new identities and construction methods for transform pairs.

Findings

Derived new identities involving Fibonacci, Bernoulli, Catalan, harmonic, and Stirling numbers.

Established relations for constructing new binomial-transform pairs.

Presented results on self-inverse sequences.

Abstract

Given any two sequences of complex numbers, we establish simple relations between their binomial convolution and the binomial convolution of their individual binomial transforms. We employ these relations to derive new identities involving Fibonacci numbers, Bernoulli numbers, Catalan numbers, harmonic numbers, odd harmonic numbers, Stirling numbers of the second kind, and binomial coefficients. In addition, we present several results which allow the construction of new binomial-transform pairs from existing ones. Many new relations concerning self-inverse sequences are also derived.