Interlacing of zeros of Laguerre polynomials of equal and consecutive degree

Abstract

We investigate interlacing properties of zeros of Laguerre polynomials $L_n^{(\alpha )}(x)$ and $L_{n+1}^{(\alpha +k)}(x),$ $\alpha >-1,$ where $n\in \mathbb{N}$ and $k\in \{1,2\}$. We prove that, in general, the zeros of these polynomials interlace partially and not fully. The sharp $t-$interval within which the zeros of two equal degree Laguerre polynomials $L_n^{(\alpha )}(x)$ and $L_n^{(\alpha +t)}(x)$ are interlacing for every $n\in \mathbb{N}$ and each $\alpha >-1$ is $0<t\le 2,$ \cite{DrMu2}, and the sharp $t-$interval within which the zeros of two consecutive degree Laguerre polynomials $L_n^{(\alpha )}(x)$ and $L_{n-1}^{(\alpha +t)}(x)$ are interlacing for every $n\in \mathbb{N}$ and each $\alpha >-1$ is $0\le t\le 2,$ \cite{DrMu1}. We derive conditions on $n\in \mathbb{N}$ and $\alpha ,$ $\alpha >-1$ that determine the partial or full interlacing of the zeros of $L_n^{(\alpha )}(x)$ and the zeros of $L_n^{(\alpha +2+k)}(x),$ $k\in \{1,2\}$. We also prove that partial interlacing holds between the zeros of $L_n^{(\alpha )}(x)$ and $L_{n-1}^{(\alpha +2+k)}(x)$ when $k\in \{1,2\},$ $n\in \mathbb{N}$ and $\alpha >-1$. Numerical illustrations of interlacing and its breakdown are provided.