Interlacing of zeros of Laguerre polynomials

Abstract: The sequence of Laguerre polynomials $\{L_{n}^{(\alpha)}(x)\} _{n=0}^\infty$ is orthogonal on $(0, \infty)$ with respect to the weight function $e^{-x} x^{\alpha},\alpha > -1$ and the real distinct positive zeros of $L_{n-1}^{(\alpha)}(x)$ and $L_{n}^{(\alpha)}(x)$ are interlacing for $\alpha >-1, n \geq 2.$ D-Muldoon (2015-2019) proved that for $\alpha >-1,$ the zeros of $L_{n-1}^{(\alpha+t)}(x)$ and $L_{n}^{(\alpha)}(x)$ are interlacing for $0 \leq t \leq 2; $ the zeros of the equal degree Laguerre polynomials $L_{n}^{(\alpha)}(x)$ and $L_{n}^{(\alpha+t)}(x)$ interlace for $0 < t \leq 2$, and the interval $0 \leq t \leq 2$ is sharp for interlacing to hold for every $n \in \mathbb{N}$. Further, the zeros of $L_{n-k}^{(\alpha+t)}(x)$ and $L_{n}^{(\alpha)}(x)$ are interlacing (in the Stieltjes sense) for $0 \leq t \leq 2k$, $1 < k < n$ and the interval $0 \leq t \leq 2k$ is sharp. At OPSFA 2019, Alan Sokal: What happens to interlacing of roots if you increase parameter and increase degree of one polynomial relative to the other? Simplest case: Are the zeros of $L_{n}^{(\alpha)}(x)$ and $L_{n+1}^{(\alpha+1)}(x)$ interlacing for $\alpha > -1$ and each $n \in \mathbb{N}$? We discuss this and related cases.

complex variables

Audience: researchers in the topic

CAvid: Complex Analysis video seminar

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