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Krawtchouk-Griffiths Systems I: Matrix Approach

We call Krawtchouk-Griffiths systems, or 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. Our approach is to work directly with matrices comprising the values of the polynomials at points of a discrete grid based on the possible counting values of the underlying multinomial distribution. The starting point for the construction of a KG-system is a generating matrix satisfying the K-condition, orthogonality with respect to the basic probability distribution associated to an individual step of the multinomial process. The variables of the polynomials corresponding to matrices may be interpreted as quantum observables in the real case, or quantum variables in the complex case. The structure of the recurrence relations for the orthogonal polynomials is presented with multiplication operators as the matrices corresponding to the quantum variables. An interesting feature is that the associated random walks correspond to the Lie algebra of the representation of symmetric tensor powers of matrices.