On variation of zeros of classical discrete orthogonal polynomials

TL;DR

This paper establishes new conditions for the monotonicity of zeros in classical discrete orthogonal polynomials, enabling a unified analysis across various polynomial families and grids.

Contribution

It introduces tractable sufficient conditions for zero monotonicity based on hypergeometric difference equations, unifying the study across multiple polynomial families.

Findings

Unified framework for zero monotonicity analysis

Applicable to diverse polynomial families and grids

Simplified criteria for monotonicity

Abstract

The purpose of this note is to establish, from the hypergeometric-type difference equation introduced by Nikiforov and Uvarov, new tractable sufficient conditions for the monotonicity with respect to a real parameter of zeros of classical discrete orthogonal polynomials. This result allows one to carry out a systematic study of the monotonicity of zeros of classical orthogonal polynomials on linear, quadratic, q-linear, and q-quadratic grids. In particular, we analyze in a simple and unified way the monotonicity of the zeros of Hahn, Charlier, Krawtchouk, Meixner, Racah, dual Hahn, q-Meixner, quantum q-Krawtchouk, q-Krawtchouk, affine q-Krawtchouk, q-Charlier, Al-Salam-Carlitz, q-Hahn, little q-Jacobi, little q-Laguerre/Wall, q-Bessel, q-Racah and dual q-Hahn polynomials.