Harmonic analysis on a local field towards addition theorems for multivariate Krawtchouk polynomials

TL;DR

This paper develops addition theorems for multivariate Krawtchouk polynomials using harmonic analysis on non-Archimedean local fields, extending previous work from the univariate case and involving spherical representation decompositions.

Contribution

It introduces a new harmonic analysis framework on local fields to derive addition theorems for multivariate Krawtchouk polynomials, generalizing Dunkl's univariate results.

Findings

Established addition theorems for multivariate Krawtchouk polynomials.

Utilized decompositions of spherical representations beyond irreducibility.

Connected harmonic analysis on local fields with polynomial addition formulas.

Abstract

We aim addition theorems for multivariate Krawtchouk polynomials, following Dunkl(1976) for 1-variate case. We work on harmonic analysis on a non-Archimedean local field, that is a group theoretic situation where these polynomials play roles of the zonal spherical functions. Unlike Dunkl's case, we use decompositions of spherical representations as not necessarily irreducible. We examine translations of zonal spherical functions, and have a kind of addition theorem for multivariate Krawtchouk polynomials.