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

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.