# Zeros of linear combinations of Laguerre polynomials from different sequences

Source: [Zeros of linear combinations of Laguerre polynomials from different sequences](<https://www.sota2.com/research/paper/68fa51bd1e209131ce976134>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-28T13:55:35.159Z

Authors: K Driver, K Jordaan

Published: 2008-12-03T14:38:42.000Z

## Abstract

We study interlacing properties of the zeros of two types of linear combinations of Laguerre polynomials with different parameters, namely $R\_n=L\_n^{\\alpha}+aL\_{n}^{\\alpha'}$ and $S\_n=L\_n^{\\alpha}+bL\_{n-1}^{\\alpha'}$. Proofs and numerical counterexamples are given in situations where the zeros of $R\_n$, and $S\_n$, respectively, interlace (or do not in general) with the zeros of $L\_k^{\\alpha}$, $L\_k^{\\alpha'}$, $k=n$ or $n-1$. The results we prove hold for continuous, as well as integral, shifts of the parameter $\\alpha$.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/0812.0730>)

[PDF](<https://arxiv.org/pdf/0812.0730.pdf>)