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

We consider the problem of computing satisfactory pairs of solutions of the differential equation for Legendre functions of non-negative integer order $μ$ and degree $-\frac12+iτ$, where $τ$ is a non-negative real parameter. Solutions of this equation are the conical functions ${\rm{P}}^μ_{-\frac12+iτ}(x)$ and ${Q}^μ_{-\frac12+iτ}(x)$, $x>-1$. 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 ${\rm{P}}^μ_{-\frac12+iτ}(x)$ for $x>1$, the function $\Re\left\{e^{-iπμ} {Q}^μ_{-\frac{1}{2}+iτ}(x) \right\}$. The proposed algorithm allows the computation of the function on a large parameter domain without requiring the use of extended precision arithmetic.

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