On parameter derivatives of the associated Legendre function of the first kind (with applications to the construction of the associated Legendre function of the second kind of integer degree and order)

Abstract

A relationship between partial derivatives of the associated Legendre function of the first kind with respect to its degree, $[\partial P_{\nu}^{m}(z)/\partial\nu]_{\nu=n}$, and to its order, $[\partial P_{n}^{\mu}(z)/\partial\mu]_{\mu=m}$, is established for $m,n\in\mathbb{N}$. This relationship is used to deduce four new closed-form representations of $[\partial P_{\nu}^{m}(z)/\partial\nu]_{\nu=n}$ from those found recently for $[\partial P_{n}^{\mu}(z)/\partial\mu]_{\mu=m}$ by the present author [R. Szmytkowski, J. Math. Chem. 46 (2009) 231]. Several new expressions for the associated Legendre function of the second kind of integer degree and order, $Q_{n}^{m}(z)$, suitable for numerical purposes, are also derived.