  This paper defines the multivariate Krawtchouk polynomials, orthogonal on the
multinomial distribution, and summarizes their properties as a review. The
multivariate Krawtchouk polynomials are symmetric functions of orthogonal sets
of functions defined on each of N multinomial trials. The dual multivariate
Krawtchouk polynomials, which also have a polynomial structure, are seen to
occur naturally as spectral orthogonal polynomials in a Karlin and McGregor
spectral representation of transition functions in a composition birth and
death process. In this Markov composition process in continuous time there are
N independent and identically distributed birth and death processes each with
support 0,1, .... The state space in the composition process is the number of
processes in the different states 0,1,... Dealing with the spectral
representation requires new extensions of the multivariate Krawtchouk
polynomials to orthogonal polynomials on a multinomial distribution with a
countable infinity of states.
