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.

翻译：本文研究具有两个相邻正则奇点 $z=0$ 和 $z=1$ 的 $2\times 2$ 线性常微分系统局部基本解之间的连接系数。我们推导了连接系数的渐近公式，该公式可用于数值计算，特别是用于确定数学物理中出现的某些谱问题的特征值。作为应用，本文提出了计算椭球波方程和球状波方程特征值的新算法。