A Formal Refutation of the Hypergeometric Parametric Extension for Reciprocal 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.
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