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

Legendre functions, conical functions, three-term recurrence relations, numerical methods for special functions
Computation of special functions, construction of tables (msc 65D20)
Other (theme 6)
Unspecified
dx.doi.org/10.1007/s11075-014-9857-5
Numerical Algorithms

Dunster, T.M, Gil, A, Segura, J, & Temme, N.M. (2015). Computation of a numerically satisfactory pair of solutions of the differential equation for conical functions of non-negative integer orders. Numerical Algorithms, 68(3), 497–509. doi:10.1007/s11075-014-9857-5