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 mu and degree , where tau is a non-negative real parameter. Solutions of this equation are the conical functions and , x >-1. An algorithm for computing a numerically satisfactory pair of solutions is already available when -1 < x < 1 (see Gil et al. SIAM J. Sci. Comput. 31(3):1716-1741, 2009, Comput. Phys. Commun. 183:794-799, 2012). In this paper, we present a stable computational scheme for a real valued numerically satisfactory companion of the function for x > 1, the function . The proposed algorithm allows the computation of the function on a large parameter domain without requiring the use of extended precision arithmetic.