Computation of a numerically satisfactory pair of solutions of the differential equation for conical functions of non-negative integer orders

Abstract

We consider the problem of computing satisfactory pairs of solutions of the differential equation for Legendre functions of non-negative integer order $\hat{I}¼$ and degree $-\frac{1}{2}+i\ddot{I}„$, where $\ddot{I}„$ is a non-negative real parameter. Solutions of this equation are the conical functions ${\mathrm{P}}^{\hat{I}}{¼}_{-\frac{1}{2}+i\ddot{I}„}(x)$ and $Q^{\hat{I}}{¼}_{-\frac{1}{2}+i\ddot{I}„}(x)$, $x>-1$. An algorithm for computing a numerically satisfactory pair of solutions is already available when $-1<x<1$ (see \cite{gil:2009:con}, \cite{gil:2012:cpc}).In this paper, we present a stable computational scheme for a real valued numerically satisfactory companion of the function ${\mathrm{P}}^{\hat{I}}{¼}_{-\frac{1}{2}+i\ddot{I}„}(x)$ for $x>1$, the function $\Re \left\{e^{-i\ddot{I}€\hat{I}¼}{Q}^{\hat{I}}{¼}_{-\frac{1}{2}+i\ddot{I}„}(x)\right\}$. The proposed algorithm allows the computation of the function on a large parameter domain without requiring the use of extended precision arithmetic.

To be published in Numerical Algoritms