On Gegenbauer polynomials and Wronskian determinants of trigonometric functions

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.