Zeros of Meixner and Krawtchouk polynomials
A Jooste, K Jordaan, F Tookos

TL;DR
This paper studies the zeros of Meixner and Krawtchouk polynomials, revealing their real and positive zeros across various parameter ranges, including non-orthogonal cases, using advanced mathematical techniques.
Contribution
It provides new results on the zero locations of Meixner and Krawtchouk polynomials beyond orthogonality conditions, including non-orthogonal and quasi-orthogonal cases.
Findings
Zeros of Krawtchouk polynomials are real and positive for certain parameters.
All zeros of Meixner polynomials are real and positive in specified parameter ranges.
Polynomials are real-rooted as parameter p approaches zero.
Abstract
We investigate the zeros of a family of hypergeometric polynomials , that are known as the Meixner polynomials for certain values of the parameters and . When , and , the polynomials , , are referred to as Krawtchouk polynomials. We prove results for the zero location of the orthogonal polynomials , and , the quasi-orthogonal polynomials , , and or as well as the non-orthogonal polynomials , and . We also show that the polynomials , are real-rooted when We use a generalised Sturmian sequence argument and the discrete orthogonality of the Krawtchouk polynomials for certain parameter values to prove…
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Taxonomy
TopicsMathematical functions and polynomials · Advanced Mathematical Identities · Polynomial and algebraic computation
