On Gegenbauer polynomials and Wronskian determinants of trigonometric functions

TL;DR

This paper generalizes a known result on the Wronskian of sine functions by expressing it in terms of Gegenbauer polynomials, providing two different proofs including recurrence relations and Darboux-Crum transformations.

Contribution

It extends the evaluation of Wronskians of sine functions to a broader set involving Gegenbauer polynomials, offering two distinct proof methods.

Findings

The Wronskian can be expressed using Gegenbauer polynomials.

Two proofs are provided: recurrence relations and Darboux-Crum transformations.

The results generalize previous work by Larsen.

Abstract

M. E. Larsen evaluated the Wronskian determinant of functions $\{\sin(mx)\}_{1\le m \le n}$. We generalize this result and compute the Wronskian of $\{\sin(mx)\}_{1\le m \le n-1}\cup \{\sin((k+n)x\} $. We show that this determinant can be expressed in terms of Gegenbauer orthogonal polynomials and we give two proofs of this result: a direct proof using recurrence relations and a less direct (but, possibly, more instructive) proof based on Darboux-Crum transformations.