A bivariate approach to realrootedness of special polynomials

TL;DR

This paper investigates the real-rootedness of special orthogonal polynomials, revealing new monotonicity and interlacing properties of their roots with respect to parameters, providing a dual perspective on orthogonality.

Contribution

It introduces a novel bivariate approach to analyze the real-rootedness and root interlacing of parameter-dependent orthogonal polynomials like Laguerre and Gegenbauer.

Findings

Proves real-rootedness of polynomials in parameter z for x in the orthogonality support.

Establishes interlacing properties of derivatives of polynomials with respect to z.

Provides a dual approach to understanding orthogonality through root behavior.

Abstract

In this paper, we exhibit new monotonicity properties of roots of families of orthogonal polynomials $P_n^{(z)}(x)$ depending polynomially on a parameter (Laguerre and Gegenbauer). By establishing that $P_n^{(z)}(x)$ are realrooted in $z$ for $x$ in the support of orthogonality, we show realrootedness in $x$ and interlacing properties of $\partial_z^kP_n^{(z)}(x)$ for $k\leq n$ and $z \geq 0$, establishing a dual approach to orthogonality.