On the connection coefficients for linear differential systems with applications to the spheroidal and ellipsoidal wave equation

This paper is concerned with the connection coefficients between the local fundamental solutions of a $2\times 2$ linear ordinary differential system with two neighboring regular singular points at $z=0$ and $z=1$. We derive an asymptotic formula for the connection coefficients which can be used for numerical calculations and, in particular, for determining the eigenvalues of some spectral problems arising in mathematical physics. As an application, new algorithms for computing the eigenvalues of the ellipsoidal wave equation and the spheroidal wave equation are presented.