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 $|\mu|$.

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SIAM Journal on Scientific Computing

Gil, A., Segura, J., & Temme, N. (2009). Computing the conical function $P^{\mu}_{-1/2+i\tau}(x)$. SIAM Journal on Scientific Computing, 31(3), 1716–1741.