Zeros of linear combinations of Laguerre polynomials from different sequences

Abstract

We study interlacing properties of the zeros of two types of linear combinations of Laguerre polynomials with different parameters, namely $R_n=L_n^{\alpha }+a{L}_n^{{\alpha }^{\prime }}$ and $S_n=L_n^{\alpha }+b{L}_{n-1}^{{\alpha }^{\prime }}$. 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 $L_k^{\alpha }$, $L_k^{{\alpha }^{\prime }}$, $k=n$ or $n-1$. The results we prove hold for continuous, as well as integral, shifts of the parameter $\alpha$.

References in corpus (1)

• When do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials?

Cited by in corpus (1)

• On linearly related orthogonal polynomials in several variables