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arXiv · 2301.00642

A bivariate approach to realrootedness of special polynomials

Aurelien Xavier Gribinski
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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.

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Aurelien Xavier Gribinski. 2022-12-30. A bivariate approach to realrootedness of special polynomials. https://arxiv.org/abs/2301.00642

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