Two Binomial Identities of Ruehr Revisited

A curious identity, proposed by Kimura and proved by Kimura, Ruehr and others, involves two definite integrals of the same continuous function f composed with the polynomial . In his proof Ruehr indicates, without giving an explicit proof, that this identity, applied to , implies two equalities involving binomial sums. Using two identities given in a book of Comtet we provide an easy explicit way of deducing these equalities from Kimura's identity between integrals. Our derivation shows a link with the incomplete beta function, the binomial distribution law, the negative binomial distribution law, and a lemma used in a proof of a very weak form of the 3x + 1 conjecture.