On Parameter Differentiation for Integral Representations of Associated Legendre Functions

TL;DR

This paper rigorously justifies differentiation under the integral sign for associated Legendre functions, enabling the evaluation of derivatives with respect to parameters at specific degrees and orders, expanding the theoretical understanding of these functions.

Contribution

It provides a rigorous justification for parameter differentiation in integral representations of associated Legendre functions, facilitating new derivative evaluations at special degrees and orders.

Findings

Derivatives of Legendre functions at half-integer degrees are evaluated.

Derivatives with respect to order at integer degrees are obtained.

Properties of a related complex function are discussed.

Abstract

For integral representations of associated Legendre functions in terms of modified Bessel functions, we establish justification for differentiation under the integral sign with respect to parameters. With this justification, derivatives for associated Legendre functions of the first and second kind with respect to the degree are evaluated at odd-half-integer degrees, for general complex-orders, and derivatives with respect to the order are evaluated at integer-orders, for general complex-degrees. We also discuss the properties of the complex function $f:{\mathbb C}\setminus\{-1,1\}\to{\mathbb C}$ given by $f(z)=z/(\sqrt{z+1}\sqrt{z-1})$.