Legendre Functions of Fractional Degree: Transformations and Evaluations

TL;DR

This paper derives transformations for Legendre functions of fractional degree, enabling their reduction to classical cases and facilitating their symbolic manipulation and evaluation in applied mathematics.

Contribution

It introduces new transformations that relate fractional degree Legendre functions to classical cases, generalizing Ramanujan's and Whipple's formulas.

Findings

Derived transformations reduce fractional degrees to classical cases.

Facilitated symbolic manipulation of Legendre and Ferrers functions.

Generalized Ramanujan's transformations and Whipple's formula.

Abstract

Associated Legendre functions of fractional degree appear in the solution of boundary value problems in wedges or in toroidal geometries, and elsewhere in applied mathematics. In the classical case when the degree is half an odd integer, they can be expressed using complete elliptic integrals. In this study, many transformations are derived, which reduce the case when the degree differs from an integer by one-third, one-fourth or one-sixth to the classical case. These transformations, or identities, facilitate the symbolic manipulation and evaluation of Legendre and Ferrers functions. They generalize both Ramanujan's transformations of elliptic integrals and Whipple's formula, which relates Legendre functions of the first and second kinds. The proofs employ algebraic coordinate transformations, specified by algebraic curves.