Associated Legendre Functions and Spherical Harmonics of Fractional Degree and Order

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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 nonclassical 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.

Original languageEnglish (US)
Pages (from-to)235-281
Number of pages47
JournalConstructive Approximation
Volume48
Issue number2
DOIs
StatePublished - Oct 1 2018

Keywords

  • Algebraic function
  • Associated Legendre function
  • Heun polynomial
  • Jacobi polynomial
  • Ladder operator
  • Solid harmonic
  • Spherical harmonic

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)
  • Computational Mathematics

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