Interlacing theorems for the zeros of some orthogonal polynomials from different sequences

doi:10.1016/j.apnum.2009.04.002

Applied Numerical Mathematics

Volume 59, Issue 8, August 2009, Pages 2015-2022

https://doi.org/10.1016/j.apnum.2009.04.002 Get rights and content

Abstract

We study the interlacing properties of the zeros of orthogonal polynomials $p_{n}$ and $r_{m}$ , $m = n$ or $n - 1$ where ${p_{n}}_{n = 1}^{\infty}$ and ${r_{m}}_{m = 1}^{\infty}$ are different sequences of orthogonal polynomials. The results obtained extend a conjecture by Askey, that the zeros of Jacobi polynomials $p_{n} = P_{n}^{(α, β)}$ and $r_{n} = P_{n}^{(γ, β)}$ interlace when $α < γ ⩽ α + 2$ , showing that the conjecture is true not only for Jacobi polynomials but also holds for Meixner, Meixner–Pollaczek, Krawtchouk and Hahn polynomials with continuously shifted parameters. Numerical examples are given to illustrate cases where the zeros do not separate each other.

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The separation of the zeros of different sequences of Hahn polynomials of the same or adjacent degree was first studied by Levit [20] in 1967, and similar interlacing results followed for Jacobi polynomials [1,5], Krawtchouk polynomials [3,11] and Meixner and Meixner–Pollaczek polynomials [11].
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¹: Research by this author is partially supported by the National Research Foundation under grant number 2054423.

²: Research by this author is partially supported by OTKA 49448.

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Interlacing theorems for the zeros of some orthogonal polynomials from different sequences

Abstract

Appl. Numer. Math.

Appl. Numer. Math.

Special Functions

The Hahn and Meixner polynomials of an imaginary argument and some of their applications

J. Phys. A: Math. Gen.

Interlacing of zeros of shifted sequences of one-parameter orthogonal polynomials

Numer. Math.

Interlacing of the zeros of Jacobi polynomials with different parameters

Numer. Algorithms

Classical and Quantum Orthogonal Polynomials in One Variable