Infinite-Variate Extensions of Krawtchouk Polynomials and Zonal Spherical Functions over a Local Field

The multivariate Krawtchouk polynomials are orthogonal polynomials for the multinomial distribution, first defined by Griffiths in 1971. We construct infinite-variate extensions of them as complete orthogonal systems of specific weighted l2-spaces. We also give realizations of our infinite-variate extensions as zonal spherical functions on groups over a non-Archimedean local field. Some typical properties of Krawtchouk polynomials like duality, orthogonality and completeness are thus shed light from the point of view of zonal spherical functions.