  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.
