Zeros of linear combinations of Laguerre polynomials from different sequences

TL;DR

This paper investigates the zero interlacing properties of specific linear combinations of Laguerre polynomials with different parameters, providing proofs and counterexamples for various parameter shifts.

Contribution

It offers new insights into the interlacing behavior of zeros of combined Laguerre polynomials with different parameters, including continuous and integral shifts.

Findings

Zeros of certain linear combinations do interlace under specific conditions.

Counterexamples show interlacing does not always hold.

Results extend to continuous and integral parameter shifts.

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 $\alpha$.