Krawtchouk-Griffiths systems II: as Bernoulli systems

Abstract

We call Krawtchouk-Griffiths systems, KG-systems, systems of multivariate polynomials orthogonal with respect to corresponding multinomial distributions. The original Krawtchouk polynomials are orthogonal with respect to a binomial distribution. Here we present a Fock space construction with raising and lowering operators. The operators of "multiplication by X" are found in terms of boson operators and corresponding recurrence relations presented. The Riccati partial differential equations for the differentiation operators, Berezin transform and associated partial differential equations are found. These features provide the specifications for a Bernoulli system as a quantization formulation of multivariate Krawtchouk polynomials.

Reproducing kernel orthogonal polynomials on the multinomial distribution

“…These are multivariate versions of the 1-dimensional expansions of Karlin and McGregor [25,26,27]. Recent representations and derivations of orthogonality of these polynomials are in Feinsilver [12,13], Genest, Vinet, and Zhedanov [15], Grunbaum and Rahman [22], Iliev [23], Mizukawa [34]. Zhou and Lange [38] show that these polynomials are eigenfunctions in classes of reversible composition Markov chains which have multinomial stationary distributions and use them to get sharp rates of convergence to stationarity.…”