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 $|\mu|$.

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.