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Failed to parse (unknown function\vu): ~(\nu - \mu + 1)P^\mu_{\vu + 1}

~=

~
e^{i \mu \pi} ~ (2\pi)^{-\frac{1}{2}} (z^2-1)^{\mu/2} ~\Gamma(\mu + \tfrac{1}{2})~\biggl\{
\int_0^\pi (z - \cos t)^{-\mu - \frac{1}{2}} \cos[(\nu + \tfrac{1}{2})t] ~dt
-\cos(\nu\pi) \int_0^\infty (z + \cosh t)^{-\mu - \frac{1}{2}} e^{-(\nu + \frac{1}{2})t} ~dt
\biggr\}

Abramowitz & Stegun (1995), p. 334, eq. (8.5.3)

NOTE: Both ~P_\nu^\mu and ~Q_\nu^\mu satisfy this same recurrence relation.

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