Interlacing of zeros of Laguerre polynomials of equal and consecutive degree

Abstract

We investigate interlacing properties of zeros of Laguerre polynomials $L_n^{\left(\alpha \right)}\left(x\right)$ and $L_{n+1}^{(\alpha +k)}\left(x\right),$ $\alpha >-1,$ 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 $L_n^{\left(\alpha \right)}\left(x\right)$ and $L_n^{(\alpha +t)}\left(x\right)$ are interlacing for every $n\in \mathbb{N}$ and each $\alpha >-1$ is $0<t\le 2,$ [Driver K, Muldoon ME. Sharp interval for interlacing of zeros of equal degree Laguerre polynomials. J Approx Theory, to appear.], and the sharp t-interval within which the zeros of two consecutive degree Laguerre polynomials $L_n^{\left(\alpha \right)}\left(x\right)$ and $L_{n-1}^{(\alpha +t)}\left(x\right)$ are interlacing for every $n\in \mathbb{N}$ and each $\alpha >-1$ is $0\le t\le 2,$ [Driver K, Muldoon ME. Common and interlacing zeros of families of Laguerre polynomials. J Approx Theory. 2015;193:89–98]. We derive conditions on $n\in \mathbb{N}$ and α, $\alpha >-1$ that determine the partial or full interlacing of the zeros of $L_n^{\left(\alpha \right)}\left(x\right)$ and the zeros of $L_n^{(\alpha +2+k)}\left(x\right),$ $k\in \{1,2\}$. We also prove that partial interlacing holds between the zeros of $L_n^{\left(\alpha \right)}\left(x\right)$ and $L_{n-1}^{(\alpha +2+k)}\left(x\right)$ when $k\in \{1,2\},$ $n\in \mathbb{N}$ and $\alpha >-1$. Numerical illustrations of interlacing and its breakdown are provided.

Keywords:

• Laguerre polynomials
• zeros
• interlacing
• three-term recurrence relation

2010 Mathematics Subject Classifications:

• 33C45
• 33C05

Acknowledgments

Jorge Arvesú and Kathy Driver wish to thank the Mathematics Department at Baylor University for hosting their visits in Fall 2019 which stimulated this research.

Figure 3. The roots of $L_3^{10}\left(x\right)$ are depicted by dots in gray and those of $L_2^{14}\left(x\right)$ are the black dots. We see that the zeros are not interlacing.

Table 5. The zeros of $L_n^{\left(\alpha \right)}\left(x\right)$ and $L_{n-1}^{(\alpha +4)}\left(x\right)$, for n = 7 and $\alpha =-1/2$.

Table 6. The zeros of $L_n^{\left(\alpha \right)}\left(x\right)$ and $L_{n-1}^{(\alpha +4)}\left(x\right)$ are interlacing when n = 6 and $\alpha =140.$

Table 7. The zeros of $L_n^{\left(\alpha \right)}\left(x\right)$ and $L_{n-1}^{(\alpha +4)}\left(x\right)$, for n = 8 and $\alpha =50.$

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research of J. Arvesú was funded by Agencia Estatal de Investigación of Spain [grant number PGC-2018-096504-B-C33]. The research of K. Driver was funded by the National Research Foundation of South Africa [grant number 115332].