The Hurwitz-type theorem for the regular Coulomb wave function via Hankel determinants

TL;DR

This paper establishes a new Hurwitz-type theorem for the zeros of the regular Coulomb wave function using Hankel determinants, providing algebraic proofs and extending classical results with novel identities.

Contribution

It derives a closed-form determinant formula for sums over Coulomb wave zeros and applies it to prove a new Hurwitz-type theorem, extending classical results to Coulomb functions.

Findings

New determinant formula for Coulomb wave zeros

Proof of a Hurwitz-type theorem for Coulomb functions

Evaluation of Hankel determinants involving Bernoulli numbers

Abstract

We derive a closed formula for the determinant of the Hankel matrix whose entries are given by sums of negative powers of the zeros of the regular Coulomb wave function. This new identity applied together with results of Grommer and Chebotarev allows us to prove a Hurwitz-type theorem about the zeros of the regular Coulomb wave function. As a particular case, we obtain a new proof of the classical Hurwitz's theorem from the theory of Bessel functions that is based on algebraic arguments. In addition, several Hankel determinants with entries given by the Rayleigh function and Bernoulli numbers are also evaluated.