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Interlacing of zeros of Laguerre polynomials of equal and consecutive degree

We investigate interlacing properties of zeros of Laguerre polynomials $ L_{n}^{(α)}(x)$ and $ L_{n+1}^{(α+k)}(x),$ $ α> -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^{(α)}(x)$ and $ L_n^{(α+t)}(x)$ are interlacing for every $n \in \mathbb{N}$ and each $ α> -1$ is $ 0 < t \leq 2,$ \cite{DrMu2}, and the sharp $t-$interval within which the zeros of two consecutive degree Laguerre polynomials $ L_n^{(α)}(x)$ and $ L_{n-1}^{(α+t)}(x)$ are interlacing for every $n \in \mathbb{N}$ and each $ α> -1$ is $ 0 \leq t \leq 2,$ \cite{DrMu1}. We derive conditions on $n \in \mathbb{N}$ and $α,$ $ α> -1$ that determine the partial or full interlacing of the zeros of $ L_n^{(α)}(x)$ and the zeros of $ L_n^{(α+ 2 + k)}(x),$ $ k \in {\{ 1,2 }\}$. We also prove that partial interlacing holds between the zeros of $ L_n^{(α)}(x)$ and $ L_{n-1}^{(α+ 2 +k )}(x)$ when $ k \in {\{ 1,2 }\},$ $n \in \mathbb{N}$ and $ α> -1$. Numerical illustrations of interlacing and its breakdown are provided.