A generalization of an identity due to Kimura and Ruehr

TL;DR

This paper generalizes a known integral identity involving a specific polynomial by introducing a family of related polynomials, expanding the scope of such identities and connecting them to Chebyshev polynomials.

Contribution

The authors extend a classical integral identity to a broader family of polynomials related to Chebyshev polynomials, demonstrating that the original identity is a special case.

Findings

Established a family of polynomial identities generalizing the original

Connected the identities to Chebyshev polynomials

Showed the original identity is a particular case of the family

Abstract

An identity stated by Kimura and proved by Ruehr, Kimura and others stipulates that for any function $f$ continuous on $[-\frac{1}{2}, \frac{3}{2}]$ one has $$ \int_{-1/2}^{3/2} f(3x^2 - 2x^3) dx = 2 \int_0^1 f(3x^2 - 2x^3) dx. $$ We prove that this equality is not an isolated example by providing a family of polynomials, related to the Tchebychev polynomials and of which $(3x^2 - 2x^3)$ is a particular case, giving rise to similar identities.