Bounds for extreme zeros of Meixner-Pollaczek polynomials

2023/12/15 by Jooste, AS, Jordaan, K.
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2312.09931

Abstract

In this paper we consider connection formulae for orthogonal polynomials in the context of Christoffel transformations for the case where a weight function, not necessarily even, is multiplied by an even function c_2k(x),k∈ N₀, to determine new lower bounds for the largest zero and upper bounds for the smallest zero of a Meixner-Pollaczek polynomial. When p_n is orthogonal with respect to a weight w(x) and g_n-m is orthogonal with respect to the weight c_2k(x)w(x), we show that k∈\0,1,…,m\ is a necessary and sufficient condition for existence of a connection formula involving a polynomial G_m-1 of degree (m-1), such that the (n-1) zeros of G_m-1g_n-m and the n zeros of p_n interlace. We analyse the new inner bounds for the extreme zeros of Meixner-Pollaczek polynomials to determine which bounds are the sharpest. We also briefly discuss bounds for the zeros of Pseudo-Jacobi polynomials.

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