Associated Legendre Functions and Spherical Harmonics of Fractional Degree and Order

Research output: Research - peer-reviewArticle

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 (Formula presented.) is generated by the ladder operators that shift the degree and order of the corresponding solid harmonics. All such representations of (Formula presented.) are shown to have a common value for each of its two Casimir invariants. The Dirac singleton representations of (Formula presented.) are included.

LanguageEnglish (US)
Pages1-47
Number of pages47
JournalConstructive Approximation
DOIs
StateAccepted/In press - Nov 13 2017

Fingerprint

Legendre function
Spherical Harmonics
Fractional
Polynomials
Monodromy
Polynomial
Family
Gauss Hypergeometric Function
Shift Operator
Jacobi Polynomials
Approximation Theory
Orthogonality
Recurrence
Paul Adrien Maurice Dirac
Lie Algebra
Harmonic
Invariant
Approximation theory
Ladders
Algebra

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

Cite this

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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 (Formula presented.) is generated by the ladder operators that shift the degree and order of the corresponding solid harmonics. All such representations of (Formula presented.) are shown to have a common value for each of its two Casimir invariants. The Dirac singleton representations of (Formula presented.) are included.",
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