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

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