Convergent Expansions for Solutions of Linear Ordinary Differential Equations Having a Simple Pole, with an Application to Associated Legendre Functions

Second‐order linear ordinary differential equations with a large parameter u are examined. Asymptotic expansions involving modified Bessel functions are applicable for the case where the coefficient function of the large parameter has a simple pole. In this paper, we examine such equations in the complex plane, and convert the asymptotic expansions into uniformly convergent series, where u appears in an inverse factorial, rather than an inverse power. Under certain mild conditions, the region of convergence containing the simple pole is unbounded. The theory is applied to obtain exact connection formulas for general solutions of the equation, and also, in a special case, to obtain convergent expansions for associated Legendre functions of complex argument and large degree.