  Diaconis and Griffiths (2014) study the multivariate Krawtchouk polynomials
orthogonal on the multinomial distribution.
  In this paper we derive the reproducing kernel orthogonal polynomials
Q_n(x,y};N,p) on the multinomial distribution which are sums of products of
orthonormal polynomials in x and y of fixed total degree n=0,1,.., N.
sum_{n=0}^N rho^nQ_n(x,y);N,p) arises naturally from a probabilistic argument.
An application to a multinomial goodness of fit test is developed, where the
chi-squared test statistic is decomposed into orthogonal components which test
the order of fit. A new duplication formula for the reproducing kernel
polynomials in terms of the 1-dimensional Krawtchouk polynomials is derived.
The duplication formula allows a Lancaster characterization of all reversible
Markov chains with a multinomial stationary distribution whose eigenvectors are
multivariate Krawtchouk polynomials and where eigenvalues are repeated within
the same total degree. The \chi^2 cutoff time, and total variation cutoff time
is investigated in such chains.
  Emphasis throughout the paper is on a probabilistic understanding of the
polynomials and their applications, particularly to Markov chains.
