Krawtchouk-Griffiths Systems II: As Bernoulli Systems

TL;DR

This paper develops a Fock space framework for multivariate Krawtchouk polynomials, connecting them to Bernoulli systems and quantum-like operator structures, expanding their mathematical and physical understanding.

Contribution

It introduces a Fock space construction with raising and lowering operators for multivariate Krawtchouk polynomials, linking them to Bernoulli systems and quantization.

Findings

Fock space construction for KG-systems

Recurrence relations for multiplication operators

Partial differential equations for differentiation operators

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.