A generalization of an identity due to Kimura and Ruehr

Jean-Paul Allouche

A generalization of an identity due to Kimura and Ruehr

Number Theory 2017-06-28 v1

Authors: Jean-Paul Allouche

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.

Keywords

integral identity Kimura and Ruehr Tchebychev polynomials polynomial family generalization

Cite

@article{arxiv.1706.08929,
  title  = {A generalization of an identity due to Kimura and Ruehr},
  author = {Jean-Paul Allouche},
  journal= {arXiv preprint arXiv:1706.08929},
  year   = {2017}
}

A generalization of an identity due to Kimura and Ruehr

Abstract

Keywords

Cite

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