# Associated Legendre Functions and Spherical Harmonics of Fractional Degree and Order

## TL;DR

This paper derives formulas and explores properties of associated Legendre functions of fractional degree and order, revealing their algebraic structure, special polynomial forms, and connections to Lie algebra representations, with implications for approximation theory.

## Contribution

It introduces new trigonometric formulas for fractional associated Legendre functions, identifies their polynomial and algebraic structures, and links them to Lie algebra representations.

## Key findings

- Expressed dihedral Legendre functions via Jacobi polynomials.
- Identified octahedral polynomial with non-classical orthogonality.
- Established Lie algebra representations with common Casimir invariants.

## Abstract

Trigonometric formulas are derived for certain families of associated Legendre functions of fractional degree and order, for use in approximation theory. These functions are algebraic, and when viewed as Gauss hypergeometric functions, belong to types classified by Schwarz, with dihedral, tetrahedral, or octahedral monodromy. The dihedral Legendre functions are expressed in terms of Jacobi polynomials. For the last two monodromy types, an underlying `octahedral' polynomial, indexed by the degree and order and having a non-classical kind of orthogonality, is identified, and recurrences for it are worked out. It is a (generalized) Heun polynomial, not a hypergeometric one. For each of these families of algebraic associated Legendre functions, a representation of the rank-2 Lie algebra so(5,C) is generated by the ladder operators that shift the degree and order of the corresponding solid harmonics. All such representations of so(5,C) are shown to have a common value for each of its two Casimir invariants. The Dirac singleton representations of so(3,2) are included.