A stable computational scheme for the conical function $P^{\mu}_{-1/2+i\tau}(x)$ for $x>-1$, real $\tau$, and $\mu\le 0$ or $\mu\in\mathbb{N}$ is presented. The scheme combines uniform asymptotic expansions for large $|\mu|$ with the application of the three-term recurrence relation on the $\mu$ index in the direction of decreasing $|\mu|$ when $x>0$. When $x<0$, the conditioning of recursion is the opposite, and conical functions can be computed in the direction of increasing $|\mu|$.

Additional Metadata
Keywords Legendre functions, conical functions, hypergeometric functions, modified Bessel functions, three-term recurrence relations, difference equations, stability of recurrence relations, numerical evaluation of special functions, asymptotic analysis
MSC Computation of special functions, construction of tables (msc 65D20)
Publisher S.I.A.M.
Journal SIAM Journal on Scientific Computing
Gil, A, Segura, J, & Temme, N.M. (2009). Computing the conical function $P^{\mu}_{-1/2+i\tau}(x)$. SIAM Journal on Scientific Computing, 31(3), 1716–1741.