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On Gegenbauer polynomials and Wronskian determinants of trigonometric functions

Minjian Yuan

TL;DR

This work generalizes Larsen's explicit Wronskian formula for the sine family by evaluating $W^{(k)}_n(x)$, the Wronskian of $\{\sin(mx)\}_{m=1}^{n-1}\cup \{\sin((n+k)x)\}$, and shows $W^{(k)}_n(x)=(-2)^{\frac{n(n-1)}{2}}\sin^{\frac{n(n+1)}{2}}(x)G(n+1)C^{(n)}_{k}(\cos x)$, where $C^{(n)}_{k}$ are Gegenbauer polynomials. The authors provide two proofs: (i) a direct recurrence-based determinant calculation and (ii) a Darboux-Crum transformation approach that connects the Wronskian to Sturm–Liouville eigenfunctions and Gegenbauer polynomials. This reveals a structural link between trigonometric Wronskians and Gegenbauer polynomials and yields a spectral interpretation in terms of diffusion operators with a $\mu\csc^2(\pi x)$ potential and killed Brownian motion. Overall, the paper extends Larsen's explicit formula, gives a clean closed form, and highlights the role of Darboux theory in determinant evaluation.

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.

On Gegenbauer polynomials and Wronskian determinants of trigonometric functions

TL;DR

This work generalizes Larsen's explicit Wronskian formula for the sine family by evaluating , the Wronskian of , and shows , where are Gegenbauer polynomials. The authors provide two proofs: (i) a direct recurrence-based determinant calculation and (ii) a Darboux-Crum transformation approach that connects the Wronskian to Sturm–Liouville eigenfunctions and Gegenbauer polynomials. This reveals a structural link between trigonometric Wronskians and Gegenbauer polynomials and yields a spectral interpretation in terms of diffusion operators with a potential and killed Brownian motion. Overall, the paper extends Larsen's explicit formula, gives a clean closed form, and highlights the role of Darboux theory in determinant evaluation.

Abstract

M. E. Larsen evaluated the Wronskian determinant of functions . We generalize this result and compute the Wronskian of . 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.
Paper Structure (3 sections, 8 theorems, 59 equations, 1 figure)

This paper contains 3 sections, 8 theorems, 59 equations, 1 figure.

Table of Contents

  1. Introduction and the main result
  2. Computing Wronskians using recurrence relations
  3. Computing Wronskians using the Darboux transformation

Key Result

Theorem 1

For integer $k\ge 0$ and $n\ge 1$ where $C^{(n)}_{k}(x)$ is the Gegenbauer polynomial.

Figures (1)

  • Figure 1: Computing $W^{(k)}_{n}$ and $\widetilde{W}^{(k)}_{n}$ via recurrence relations \ref{['eqn:Wnk_recurrence']} and \ref{['eqn:Cnk_recurrence']}.

Theorems & Definitions (11)

  • Theorem 1
  • Proposition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Proposition 2
  • Theorem 2: Crum_1955Gomez_2020
  • ...and 1 more