# Orthogonality of the Ferrers' Associated Legendre Functions of the Second Kind with Imaginary Argument

## TL;DR

This paper investigates the orthogonality properties of Ferrers' associated Legendre functions of the second kind with imaginary arguments, establishing conditions for their square integrability and deriving their orthogonality relations.

## Contribution

It provides new theoretical results on the orthogonality and integrability conditions of these special functions with imaginary arguments.

## Key findings

- Derived conditions for square integrability of the functions.
- Proved the orthogonality relations for the functions.
- Identified parameter ranges where the functions form an orthogonal set.

## Abstract

In this work we study the associated Legendre functions of the second kind with a purely imaginary argument $Q^k_\ell(\mathbb{i}\, x)$. We derive the conditions under which they provide a set of square integrable functions when $x \in \mathbb{R}$ and we prove the relevant orthogonality relation that they satisfy.