Two exercises of Comtet and two identities of Ruehr

TL;DR

This paper explores integral identities related to a function involving cubic polynomials, deriving binomial sum identities through combinatorial and probabilistic methods, and connecting them to special functions and conjectures.

Contribution

It provides an explicit derivation of binomial sum identities from integral equalities using combinatorial and probabilistic techniques, linking to special functions and conjectures.

Findings

Derived binomial sum identities from integral equalities.

Connected identities to incomplete beta function and probability distributions.

Linked results to the (3x+1)-conjecture proof techniques.

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.