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<math>~(\nu - \mu + 1)P^\mu_{\vu + 1}</math>
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<math>~(\nu - \mu + 1)P^\mu_{\nu + 1} (z)</math>
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<math>~
<math>~
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e^{i \mu \pi} ~ (2\pi)^{-\frac{1}{2}} (z^2-1)^{\mu/2} ~\Gamma(\mu + \tfrac{1}{2})~\biggl\{
+
(2\nu + 1)z P_\nu^\mu(z) - (\nu + \mu)P^\mu_{\nu-1}(z)
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\int_0^\pi (z - \cos t)^{-\mu - \frac{1}{2}} \cos[(\nu + \tfrac{1}{2})t] ~dt
+
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-\cos(\nu\pi) \int_0^\infty (z + \cosh t)^{-\mu - \frac{1}{2}} e^{-(\nu + \frac{1}{2})t} ~dt
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\biggr\}
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</math>
</math>
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NOTE:  Both <math>~P_\nu^\mu</math> and <math>~Q_\nu^\mu</math> satisfy this same recurrence relation.
+
NOTE:  <math>~Q_\nu^\mu</math>, as well as <math>~P_\nu^\mu</math>, satisfies this same recurrence relation.
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Revision as of 16:53, 13 June 2018

~(\nu - \mu + 1)P^\mu_{\nu + 1} (z)

~=

~
(2\nu + 1)z P_\nu^\mu(z) - (\nu + \mu)P^\mu_{\nu-1}(z)

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

NOTE: ~Q_\nu^\mu, as well as ~P_\nu^\mu, satisfies this same recurrence relation.

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