Double sums involving binomial coefficients and special numbers

TL;DR

This paper introduces an elementary method for evaluating double sums with incomplete binomial inner sums, applying it to sums involving harmonic, Fibonacci, Stirling, and r-Stirling numbers.

Contribution

It presents a new elementary approach for double sums with incomplete binomial sums and extends it to sums involving various special numbers.

Findings

Derived identities for double sums involving harmonic, Fibonacci, Stirling, and r-Stirling numbers.

Provided a unified elementary framework for these sums.

Enhanced understanding of relationships between binomial sums and special numbers.

Abstract

In this paper, we find an elementary approach for double sums where the inner sum is binomial but incomplete. We apply our core identity and its relatives to double sums involving famous numbers such as harmonic numbers, Fibonacci numbers, Stirling numbers and $r$-Stirling numbers of the second kind.