An introduction to multivariate Krawtchouk polynomials and their applications

TL;DR

This paper introduces multivariate Krawtchouk polynomials, exploring their properties, applications in Markov chains, and their relation to multinomial distributions, providing foundational knowledge for further research in probabilistic and combinatorial contexts.

Contribution

It offers a comprehensive introduction to multivariate Krawtchouk polynomials, detailing their properties, applications in Markov chains, and connections to multinomial distributions.

Findings

Explicit diagonalization of Markov chains using these polynomials

Characterization of bivariate multinomial distributions

Foundational properties of multivariate Krawtchouk polynomials

Abstract

Orthogonal polynomials for the multinomial distribution m(x, p) of N balls dropped into d boxes (box i has probability p(i)) are called multivariate Krawtchouk polynomials. This paper gives an introduction to their properties, collections of natural Markov chains which they explicitly diagonalize and associated bivariate multinomial distributions.