Complex zeros of Bessel function derivatives and associated orthogonal polynomials

TL;DR

This paper introduces orthogonal polynomials linked to the zeros of Bessel function derivatives, providing new formulas and analyzing the zeros' distribution depending on the order.

Contribution

It develops a new class of orthogonal polynomials associated with Bessel derivative zeros and explores their properties and implications for zero distribution.

Findings

Derived a formula for the Hankel determinant involving Rayleigh-type sums.

Established recurrence and orthogonality properties of the new polynomials.

Extended Hurwitz-type theorem for zeros of Bessel derivatives.

Abstract

We introduce a sequence of orthogonal polynomials whose associated moments are the Rayleigh-type sums, involving the zeros of the Bessel derivative $J_\nu'$ of order $\nu$. We also discuss the fundamental properties of those polynomials such as recurrence, orthogonality, etc. Consequently, we obtain a formula for the Hankel determinant, elements of which are chosen as the aforementioned Rayleigh-type sums. As an application, we complete the Hurwitz-type theorem for $J_\nu'$, which deals with the number of complex zeros of $J_\nu'$ depending on the range of $\nu$.