A Formal Refutation of the Hypergeometric Parametric Extension for Reciprocal Binomial Sums

TL;DR

This paper rigorously disproves a claimed hypergeometric closed-form for a class of reciprocal binomial sums, clarifying the limitations of a recent systematic evaluation approach.

Contribution

It provides a formal proof that the proposed parametric extension for reciprocal binomial sums is incorrect, correcting the literature.

Findings

The hypergeometric identity proposed by Pain is false.

Logical and symbolic analysis confirms the non-existence of the claimed closed-form.

The work clarifies the boundaries of hypergeometric methods for binomial sums.

Abstract

Recent work by Pain [1] proposed a systematic approach to evaluating binomial sums involving reciprocals of binomial coefficients via Beta integrals. In particular, a parametric extension (Proposition 6.1) was introduced and claimed to admit a closed-form representation in terms of a terminating 2F1 hypergeometric function. Through a combination of internal logical consistency checks, integral derivation analysis, and exact symbolic computation, we definitively prove that this parametric identity is false.