Associated Legendre functions of half-odd degree and arguments larger than one, also known as toroidal harmonics, appear in the solution of Dirichlet problems with toroidal symmetry. It is shown how the use of series expansions, continued fractions and uniform asymptotic expansions, together with the application of recurrence relations over degrees and orders, permits the evaluation of the whole set of toroidal functions for a wide range of arguments, orders and degrees. In particular, we provide a suitable uniform asymptotic expansion for $P_{nu^{m(x)$ (for large $m$) which fills the gap left by previous methods.

Classical hypergeometric functions, ${}_2F_1$ (msc 33C05), Computation of special functions, construction of tables (msc 65D20), Asymptotic approximations, asymptotic expansions (steepest descent, etc.) (msc 41A60)
CWI
Modelling, Analysis and Simulation [MAS]
Computational Dynamics

Gil, A, Segura, J, & Temme, N.M. (2000). Computing toroidal functions for wide ranges of the parameters. Modelling, Analysis and Simulation [MAS]. CWI.