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On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). (English) Zbl 05969393
Summary: In our recent works (R. Szmytkowski, J. Phys. A 39:15147, 2006; corrigendum: 40:7819, 2007; addendum: 40:14887, 2007), we have investigated the derivative of the Legendre function of the first kind, ${P}_{\nu }\left(z\right)$, with respect to its degree $\nu$. In the present work, we extend these studies and construct several representations of the derivative of the associated Legendre function of the first kind, ${P}_{\nu }^{±m}\left(z\right)$, with respect to the degree $\nu$, for $m\in ℕ$. At first, we establish several contour-integral representations of $\partial {P}_{\nu }^{±m}\left(z\right)/\partial \nu$. They are then used to derive Rodrigues-type formulas for ${\left[\partial {P}_{\nu }^{±m}\left(z\right)/\partial \nu \right]}_{\nu =n}$ with $n\in ℕ$. Next, some closed-form expressions for ${\left[\partial {P}_{\nu }^{±m}\left(z\right)/\partial \nu \right]}_{\nu =n}$ are obtained. These results are applied to find several representations, both explicit and of the Rodrigues type, for the associated Legendre function of the second kind of integer degree and order, ${Q}_{n}^{±m}\left(z\right)$; the explicit representations are suitable for use for numerical purposes in various regions of the complex $z$-plane. Finally, the derivatives ${\left[{\partial }^{2}{P}_{\nu }^{m}\left(z\right)/\partial {\nu }^{2}\right]}_{\nu =n},{\left[\partial {Q}_{\nu }^{m}\left(z\right)/\partial \nu \right]}_{\nu =n}$ and ${\left[\partial {Q}_{\nu }^{m}\left(z\right)/\partial \nu \right]}_{\nu =-n-1}$, all with $m>n$, are evaluated in terms of ${\left[\partial {P}_{\nu }^{-m}\left(±z\right)/\partial \nu \right]}_{\nu =n}$. The present paper is a complementary to a recent one (R. Szmytkowski, J. Math. Chem 46:231, 2009), in which the derivative $\partial {P}_{n}^{\mu }\left(z\right)/\partial \mu$ has been investigated.
##### MSC:
 92E Chemistry