Double sums associated with binomial transforms

TL;DR

This paper explores new identities involving double sums with binomial and incomplete sums, connecting them to special numbers, polynomials, and hyperbolic functions, expanding the mathematical understanding of these sums.

Contribution

It introduces novel results for double sums related to binomial transforms and derives new identities involving special numbers and polynomials.

Findings

New identities for double sums with binomial transforms

Connections to Bernoulli, Fibonacci, harmonic, Catalan, and Stirling numbers

Inclusion of hyperbolic function sums

Abstract

In this paper, we continue our investigation of double sums where the inner sum is binomial but incomplete. We prove many new results for these types of double sums associated with binomial transform pairs. As applications we deduce new identities for double sums involving special numbers like Bernoulli numbers, Fibonacci numbers, harmonic numbers, Catalan numbers and Stirling numbers of the second kind. We also consider families of polynomials like Fibonacci polynomials, Chebyshev polynomials, Bernoulli polynomials, and others. Finally, we state new double sums involving hyperbolic functions.