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 $[-\frac{1}{2}, \frac{3}{2}]$ one has $$ \int_{-1/2}^{3/2} f(3x^2 - 2x^3) dx = 2 \int_0^1 f(3x^2 - 2x^3) dx. $$ In his proof Ruehr indicates, without giving an explicit proof, that this identity, applied to $f(t) = t^n$, implies two identities involving binomial sums, namely (after correction of a misprint) $$ \sum_{0 \leq j \leq n} 3^j {3n-j \choose 2n} = \sum_{0 \leq j \leq 2n} (-3)^j {3n-j \choose n} \ \ \mbox{and} \ \ \sum_{0 \leq j \leq n} 2^j {3n+1 \choose n-j} = \sum_{0 \leq j \leq 2n} (-4)^j {3n+1 \choose n+1+j}. $$ 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.