### 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.

Language | English (US) |
---|---|

Pages | 1-47 |

Number of pages | 47 |

Journal | Constructive Approximation |

DOIs | |

State | Accepted/In press - Nov 13 2017 |

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### 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

**Associated Legendre Functions and Spherical Harmonics of Fractional Degree and Order.** / Maier, Robert S.

Research output: Research - peer-review › Article

}

TY - JOUR

T1 - Associated Legendre Functions and Spherical Harmonics of Fractional Degree and Order

AU - Maier,Robert S.

PY - 2017/11/13

Y1 - 2017/11/13

N2 - 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.

AB - 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.

KW - Algebraic function

KW - Associated Legendre function

KW - Heun polynomial

KW - Jacobi polynomial

KW - Ladder operator

KW - Solid harmonic

KW - Spherical harmonic

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UR - http://www.scopus.com/inward/citedby.url?scp=85033558415&partnerID=8YFLogxK

U2 - 10.1007/s00365-017-9403-5

DO - 10.1007/s00365-017-9403-5

M3 - Article

SP - 1

EP - 47

JO - Constructive Approximation

T2 - Constructive Approximation

JF - Constructive Approximation

SN - 0176-4276

ER -