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Higher Monotonicity Properties for Zeros of Certain Sturm-Liouville Functions
Version 1
: Received: 22 May 2023 / Approved: 24 May 2023 / Online: 24 May 2023 (01:54:22 CEST)
A peer-reviewed article of this Preprint also exists.
Tsai, T.-M. Higher Monotonicity Properties for Zeros of Certain Sturm-Liouville Functions. Mathematics 2023, 11, 2787. Tsai, T.-M. Higher Monotonicity Properties for Zeros of Certain Sturm-Liouville Functions. Mathematics 2023, 11, 2787.
Abstract
We consider the second order differential equation y'' + ω^(2)ρ(x)y = 0 where ω is a positive
parameter. The principal concern here is to find conditions on the function
ρ^(−1/2)(x), which ensure that the consecutive differences of sequences constructed
from the zeros of a nontrivial solution of the equation are regular in sign for ω
sufficiently large. In particular, if c_(νk)(α) denotes the k-th positive zero of the general
Bessel (cylinder) function C_(ν)(x; α) = J_(ν)(x) cos α−Y_(ν)(x) sin α of order ν, and if |ν| < 1/2,
we prove that
(−1)^(m)∆^(m+2)c_(νk)(α) > 0 (m = 0, 1, 2, ...; k = 1, 2, ...),
where ∆a_(k) = a_(k+1) − a_(k). This type of inequalities was conjectured by Lorch and Szego in 1963.
We also show that the differences of the zeros of various orthogonal polynomials with higher
degrees possess the sign-regularity.
Keywords
Sturm-Liouville equations; differences; zeros; higher monotonicity; Bessel functions; orthogonal polynomials
Subject
Computer Science and Mathematics, Analysis
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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