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

TL;DR

This paper derives formulas for the derivatives of associated Legendre functions of the first kind with respect to their order, especially at integer orders, and applies these results to express related functions explicitly.

Contribution

It provides new explicit formulas for derivatives of associated Legendre functions at integer orders, extending classical representations to associated Legendre functions of the second kind.

Findings

Explicit formulas for derivatives of $P_{n}^{ u}(z)$ at $ u= ext{integer}$.

Generalized representations of $Q_{n}^{ ext{m}}(z)$ based on derivatives.

Evaluation of second derivatives and derivatives of related functions at integer orders.

Abstract

The derivative of the associated Legendre function of the first kind of integer degree with respect to its order, $\partial P_{n}^{\mu}(z)/\partial\mu$, is studied. After deriving and investigating general formulas for $\mu$ arbitrary complex, a detailed discussion of $[\partial P_{n}^{\mu}(z)/\partial\mu]_{\mu=\pm m}$, 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 $[\partial^{2} P_{n}^{\mu}(z)/\partial\mu^{2}]_{\mu=m}$, $[\partial Q_{n}^{\mu}(z)/\partial\mu]_{\mu=m}$ and $[\partial Q_{-n-1}^{\mu}(z)/\partial\mu]_{\mu=m}$, all with $m>n$, are also evaluated.