On Perfectness of Systems of Weights Satisfying Pearson’s Equation with Nonstandard Parameters

Abstract

Measures generating classical orthogonal polynomials are determined by Pearson’s equation, whose parameters usually provide the positivity of the measures. The case of general complex parameters (nonstandard) is also of interest; the non-Hermitian orthogonality with respect to (now complex-valued) measures is considered on curves in C. Some applications lead to multiple orthogonality with respect to a number of such measures. For a system of r orthogonality measures, the perfectness is an important property: in particular, it implies the uniqueness for the whole family of corresponding multiple orthogonal polynomials and the (r+2)-term recurrence relations. In this paper, we introduce a unified approach which allows to prove the perfectness of the systems of complex measures satisfying Pearson’s equation with nonstandard parameters. We also study the polynomials satisfying multiple orthogonality relations with respect to a system of discrete measures. The well-studied families of multiple Charlier, Krawtchouk, Meixner and Hahn polynomials correspond to the systems of measures defined by the difference Pearson’s equation with standard real parameters. Using the same approach, we verify the perfectness of such systems for general parameters. For some values of the parameters, discrete measures should be replaced with the continuous measures with non-real supports.