Zeros of a cross-product of the Coulomb wave and Tricomi hypergeometric functions

TL;DR

This paper proves that the positive zeros of a cross-product of Coulomb wave and Tricomi hypergeometric functions increase with the order, implying eigenvalues of a related boundary value problem also increase with dimension.

Contribution

It establishes the monotonicity of zeros of a specific cross-product of special functions, linking it to eigenvalue behavior in boundary value problems.

Findings

Zeros of the cross-product are increasing with respect to order.

Eigenvalues of the boundary value problem increase with dimension.

Provides mathematical insight into special functions and spectral theory.

Abstract

Motivated by a problem related to conditions for the existence of clines in genetics, in this note our aim is to show that the positive zeros of a cross-product of the regular Coulomb wave function and the Tricomi hypergeometric function are increasing with respect to the order. In particular, this implies that the eigenvalues of a boundary value problem are increasing with the dimension.