Continuous-Discrete Duality Formulas for Combinatorial Numbers and Their Applications to Higher-Order Series Summations
Year: 2026
Type: preprint
Abstract: This paper investigates the continuous-discrete duality of combinatorial numbers, focusing on finite sums involving inverses and higher-order powers of binomial coefficients. By employing integral representations and parameter extensions, we establish a closed-form expression for the fundamental reciprocal binomial sum, transforming a discrete summation into a continuous product and a Beta integral. Furthermore, we generalize this framework to alternating higher-order sums, connecting them with gamma functions, Hurwitz zeta functions, and higher-order polygamma functions. The proposed duality framework provides a rigorous and insightful analytical pathway for evaluating complex finite series without relying on experimental simulations.
Author: Zhang, Jincheng
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7171592616