Orthogonality of the Ferrers' associated Legendre functions of the second kind with imaginary argument

ABSTRACT

In this work we study the associated Legendre functions of the second kind with a purely imaginary argument $Q_{\ell }^k\left(\mathbb{i}x\right)$. 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.

KEYWORDS:

• Associated Legendre functions
• orthogonal functions
• square integrability

AMS SUBJECT CLASSIFICATION:

• 42C05
• 33C05
• 33C45

Disclosure statement

No potential conflict of interest was reported by the author.

ORCID

N. Dimakis http://orcid.org/0000-0002-1307-1073.

Additional information

Funding

The author acknowledges financial support by Comisión Nacional de Investigación Científica y Tecnológica; FONDECYT postdoctoral grant no. 3150016.