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)

TL;DR

This paper establishes a relationship between derivatives of the associated Legendre function of the first kind with respect to degree and order, deriving new closed-form expressions and formulas for the second kind for improved numerical computation.

Contribution

It introduces a novel relationship between derivatives of Legendre functions with respect to degree and order, and derives new explicit formulas for the second kind of Legendre functions.

Findings

New closed-form representations of derivatives of Legendre functions with respect to degree.

Explicit formulas for associated Legendre functions of the second kind for numerical use.

Relationships connecting derivatives with respect to degree and order of Legendre functions.

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.