RT Journal Article
T1 Zeros of Jacobi and ultraspherical polynomials
A1 Arvesú Carballo, Jorge
A1 Driver, K.
A1 Littlejohn ., Lance Lee
AB Suppose {P-n((alpha,beta))(x)}(n=0)(infinity) is a sequence of Jacobi polynomials with alpha, beta > -1. We discuss special cases of a question raised by Alan Sokal at OPSFA in 2019, namely, whether the zeros of P-n((alpha,beta)) (x) and P-n+k((alpha+t,beta+s)(x)) are interlacing if s, t > 0 and k is an element of N. We consider two cases of this question for Jacobi polynomials of consecutive degree and prove that the zeros of P-n((alpha,beta)) (x) and P-n+1((alpha,beta+1)) (x), alpha > -1, beta > 0, n is an element of N, are partially, but in general not fully, interlacing depending on the values of alpha, beta and n. A similar result holds for the extent to which interlacing holds between the zeros of P-n((alpha,beta)) (x) and P-n+1((alpha+1,beta+1)) (x), alpha > -1, beta > -1. It is known that the zeros of the equal degree Jacobi polynomials P-n((alpha,beta)) (x) and P-n((alpha-t,beta+s)) (x) are interlacing for alpha - t > -1, beta > -1, 0 <= t, s <= 2. We prove that partial, but in general not full, interlacing of zeros holds between the zeros of P-n((alpha,beta)) (x) and P-n((alpha+1,beta+1)) (x), when alpha > -1, beta > -1. We provide numerical examples that confirm that the results we prove cannot be strengthened in general. The symmetric case alpha = beta = lambda - 1/2 of the Jacobi polynomials is also considered. We prove that the zeros of the ultraspherical polynomials C-n((lambda))(x) and C-n+1((lambda+1)) (x), lambda > -1/2, are partially, but in general not fully, interlacing. The interlacing of the zeros of the equal degree ultraspherical polynomials C-n((lambda)) (x) and C-n((lambda+3)) (x), lambda > -1/2, is also discussed.
PB Springer
SN 1382-4090
YR 2023
FD 2023-06
LK https://hdl.handle.net/10016/39849
UL https://hdl.handle.net/10016/39849
LA eng
NO The research of J. Arvesú was funded by Agencia Estatal de Investigación of Spain, Grant Number PGC-2018-096504-B-C33. The research of K. Driver was funded by the National Research Foundation of South Africa, Grant Number 115332.
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RD 30 jun. 2024