Double sums involving binomial coefficients and special numbers
Kunle Adegoke, Robert Frontczak, Karol Gryszka
TL;DR
The paper develops an elementary framework for evaluating double sums with an incomplete inner binomial sum by proving a central identity $\displaystyle \sum_{k=0}^n x^k \left(\sum_{j=0}^k \binom{n}{j} y^j\right) = \frac{1}{1-x}\left((1+xy)^n - x^{n+1}(1+y)^n\right)$ and exploring its variations. This core result is then applied to derive new closed forms for sums involving Fibonacci and Lucas numbers, harmonic numbers, and Stirling numbers (including $r$-Stirling numbers) as well as to generalize to $m$-step sequences, Beta-integral based harmonic-number identities, and Horadam-type sequences. The work yields parity-dependent and parameterized identities, linking binomial sums to classical sequences and providing a versatile toolkit for combinatorial and number-theoretic sum evaluations. Overall, the paper offers a unified, elementary method that produces a broad collection of new double-sum identities with potential applications in analytic combinatorics and related fields.
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.