Derivatives with respect to the degree and order of associated Legendre functions for $|z|>1$ using modified Bessel functions

TL;DR

This paper derives formulas for the derivatives of associated Legendre functions with respect to their degree and order for arguments greater than one, utilizing modified Bessel functions and integral representations.

Contribution

It introduces new expressions for derivatives of associated Legendre functions with respect to degree and order for complex parameters, expanding existing mathematical tools.

Findings

Derived derivatives of associated Legendre functions at specific degrees and orders.

Provided integral representations for these derivatives.

Applied modified Bessel functions to evaluate derivatives for |z|>1.

Abstract

Expressions for the derivatives with respect to order of modified Bessel functions evaluated at integer orders and certain integral representations of associated Legendre functions with modulus argument greater than unity are used to compute derivatives of the associated Legendre functions with respect to their parameters. For the associated Legendre functions of the first and second kind, derivatives 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.