On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order

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dc.creatorSzmytkowski, Radoslaw
dc.date2008-03-27
dc.date.accessioned2026-07-07T09:28:51Z
dc.date.available2026-07-07T09:28:51Z
dc.descriptionThe derivative of the associated Legendre function of the first kind of integer degree with respect to its order, $\partial P_{n}^μ(z)/\partialμ$, is studied. After deriving and investigating general formulas for $μ$ arbitrary complex, a detailed discussion of $[\partial P_{n}^μ(z)/\partialμ]_{μ=\pm m}$, where $m$ is a non-negative integer, is carried out. The results are applied to obtain several explicit expressions for the associated Legendre function of the second kind of integer degree and order, $Q_{n}^{\pm m}(z)$. In particular, we arrive at formulas which generalize to the case of $Q_{n}^{\pm m}(z)$ ($0\leqslant m\leqslant n$) the well-known Christoffel's representation of the Legendre function of the second kind, $Q_{n}(z)$. The derivatives $[\partial^{2} P_{n}^μ(z)/\partialμ^{2}]_{μ=m}$, $[\partial Q_{n}^μ(z)/\partialμ]_{μ=m}$ and $[\partial Q_{-n-1}^μ(z)/\partialμ]_{μ=m}$, all with $m>n$, are also evaluated.
dc.descriptionLaTeX, 22 pages
dc.identifierhttps://arxiv.org/abs/0803.3993
dc.identifierhttp://arxiv.org/abs/0803.3993
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/157572
dc.subjectClassical Analysis and ODEs
dc.subjectMathematical Physics
dc.subject33C45; 33C05
dc.titleOn the derivative of the associated Legendre function of the first kind of integer degree with respect to its order
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