Derivations and identities for Kravchuk polynomials

TL;DR

This paper introduces Kravchuk derivations and shows how their kernels generate polynomial identities for Kravchuk polynomials, linking algebraic derivations to polynomial identities.

Contribution

It establishes a novel connection between Kravchuk derivations and polynomial identities, expanding understanding of Kravchuk polynomials' algebraic properties.

Findings

Kernel elements of Kravchuk derivations produce polynomial identities.

Kernel elements of Weitzenböck derivations also yield identities.

Intertwining maps between derivations are described.

Abstract

We introduce the notion of Kravchuk derivations of the polynomial algebra. We prove that any element of the kernel of the derivation gives a polynomial identity satisfied by the Kravchuk polynomials. Also, we prove that any kernel element of the basic Weitzenb\"ok derivations yields a polynomial identity satisfied by the Kravchuk polynomials. We describe the corresponding intertwining maps.