Abstract
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.
Highlights
Griffiths [1] and Diaconis and Griffiths [2] construct multivariate Krawtchouk polynomials orthogonal on the multinomial distribution and study their properties
The approach of Diaconis and Griffiths [2] is probabilistic and directed to Markov chain applications; the approach of Iliev [5] is via Lie groups; and the physics approach of Genest et al [3] is as matrix elements of group representations on oscillator states
Xu [7] studies discrete multivariate orthogonal polynomials, which have a triangular construction of products of one-dimensional orthogonal polynomials
Summary
Griffiths [1] and Diaconis and Griffiths [2] construct multivariate Krawtchouk polynomials orthogonal on the multinomial distribution and study their properties. A number of classical birth and death processes have a spectral expansion where the orthogonal polynomials are constructed from the Meixner class This class has a generating function of the form:. This paper defines the multivariate Krawtchouk polynomials, summarizes their properties, considers how they are found in spectral expansions of composition birth and death processes It is partly a review of these polynomials and is self-contained. These polynomials occur naturally as eigenfunctions in composition birth and death processes in a Karlin and McGregor spectral expansion in Theorem 9. Theorem 11 gives an interesting identity for these spectral polynomials in composition birth and death processes when the spectral polynomials in the individual processes belong to the Meixner class
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