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Zeros of linear combinations of Laguerre polynomials from different sequences

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K Driver, K Jordaan

Classical Analysis and ODEs · Submitted 3 Dec 2008

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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 $n-1$.

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arXiv ID0812.0730 (version 1)
Pith IDYJO5N5EI, Pith's permanent record of this paper and its review
CitationK Driver, K Jordaan. "Zeros of linear combinations of Laguerre polynomials from different sequences." arXiv:0812.0730 (2008). https://pith.science/paper/YJO5N5EI
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,123 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.48 · human participants 0.01 · controlled comparison 0.08 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.14 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; .

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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#sentencerolethe authors' own workstates the main conclusionstrength
s0 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 $S_n=L_n^{\alpha}+bL_{n-1}^{\alpha'}$. contribution 1.00 0.88 0.14 asserted 0.93
s1 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 $n-1$. method 0.46 0.95 0.26 asserted 0.98
s2 The results we prove hold for continuous, as well as integral, shifts of the parameter $\alpha$. result 0.93 0.86 0.13 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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The paper
Original 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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