Zeros of orthogonal little q-Jacobi polynomials: interlacing and monotonicity

TL;DR

This paper studies the zeros of little q-Jacobi polynomials, establishing their interlacing and monotonicity properties, and introduces the logarithmic mesh as a key tool for analyzing zero spacing and polynomial family classifications.

Contribution

It provides new interlacing relations, monotonicity results with respect to parameters, and structural decompositions for q-hypergeometric polynomial families, including classical limits.

Findings

Zeros exhibit strong interlacing properties

Zeros obey natural monotonicity rules with parameters

Logarithmic mesh quantifies zero spacing

Abstract

We investigate the distribution of zeros of the little q-Jacobi polynomials and related q-hypergeometric families. We prove that the zeros of these orthogonal polynomials exhibit strong interlacing properties and obey natural monotonicity rules with respect to the parameters. A key tool in our approach is the logarithmic mesh, which quantifies the relative spacing of the positive real zeros and allows us to classify families of polynomials with prescribed interlacing patterns. Our results include new interlacing relations, monotonicity with respect to parameters, and structural decompositions in non-orthogonal regimes. Several classical families of q-hypergeometric polynomials, including q-Bessel and Stieltjes-Wigert polynomials, are treated as limit cases. The methods rely on a combination of classical orthogonality theory and q-difference equations.