Reciprocal binomial sums via Beta integrals

TL;DR

This paper introduces a systematic Beta integral-based method for evaluating binomial sums involving reciprocals, connecting combinatorics with hypergeometric functions and enabling explicit formulas and identities.

Contribution

It provides a fully explicit, integral-based framework for binomial sums involving reciprocals, extending classical identities and linking to hypergeometric functions.

Findings

Derived classical identities like Frisch's formula using Beta integrals.

Established connections between binomial sums and hypergeometric series such as ${}_2F_1$ and ${}_3F_2$.

Produced explicit finite expansions for symbolic and numerical computation.

Abstract

We develop a systematic and fully explicit approach to the evaluation of binomial sums involving reciprocals of binomial coefficients based on Beta integral techniques. Starting from a simple integral representation, we provide a derivation of classical identities, including Frisch's formula, with all intermediate transformations rigorously justified. This framework naturally extends to parametric sums, yielding integral representations that lead to closed forms in terms of hypergeometric functions. In particular, we establish connections with terminating ${}_2F_1$ and generalized ${}_3F_2$ series, thereby linking discrete combinatorial sums with the analytic theory of special functions. We further derive explicit finite expansions suitable for symbolic and numerical computation, as well as higher-order extensions involving Pochhammer symbols. In addition, we present new families of identities, including shifted reciprocal sums and weighted sums involving powers of the summation index, which admit unified hypergeometric representations. Overall, the Beta integral method provides a versatile and unifying framework bridging combinatorial identities, integral representations, and hypergeometric analysis, and opens the way to further generalizations in combinatorics and special function theory.