Two exercises of Comtet and two identities of Ruehr
Abstract
A question proposed by Kimura and proved by Ruehr, Kimura and others in 1980 states that for any function f continuous on [-12, 32] one has ∫-1/23/2 f(3x2 - 2x3) dx = 2 ∫01 f(3x2 - 2x3) dx. In his proof Ruehr indicates, without giving an explicit proof, that this identity, applied to f(t) = tn, implies two identities involving binomial sums, namely (after correction of a misprint) Σ0 ≤ j ≤ n 3j 3n-j 2n = Σ0 ≤ j ≤ 2n (-3)j 3n-j n \ \ and \ \ Σ0 ≤ j ≤ n 2j 3n+1 n-j = Σ0 ≤ j ≤ 2n (-4)j 3n+1 n+1+j. Using two identities given in a book of Comtet we provide an easy explicit way of deducing these identities from the above equality 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.