Legendre functions of fractional degree: transformations and evaluations

Maier, Robert S.

doi:10.1098/rspa.2016.0097

Cited by 7 publications

(10 citation statements)

References 39 publications

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“…(A2) converges uniformly for any finite domain of ν ∈ C, see [79] (p.68) for a similar statement for the hypergeometric function. Many properties of the Legendre function are discussed in the classic textbook by Hobson [80], while their applications in electric potential problems are summarized in [81] (see also [82,83]). Here we reproduce some properties that are relevant for our work.…”

Section: Discussionmentioning

confidence: 99%

Reversible reactions controlled by surface diffusion on a sphere

Grebenkov

2019

The Journal of Chemical Physics

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We study diffusion of particles on the surface of a sphere toward a partially reactive circular target with partly reversible binding kinetics. We solve the coupled diffusion-reaction equations and obtain the exact expressions for the time-dependent concentration of particles and the total diffusive flux. Explicit asymptotic formulas are derived in the small target limit. This study reveals the strong effects of reversible binding kinetics onto diffusion-mediated reactions that may be relevant for many biochemical reactions on cell membranes.

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Section: Discussionmentioning

confidence: 99%

Reversible reactions controlled by surface diffusion on a sphere

Grebenkov

2019

The Journal of Chemical Physics

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“…The results on algebraicity are consequences of Schwarz's classification of the algebraic cases of the Gauss function 2 F 1 , and the explicit examples of algebraic generating functions came from recently developed closed-form expressions for certain algebraic 2 F 1 's with octahedral and tetrahedral monodromy; i.e., octahedral and tetrahedral associated Legendre functions. 13 This is because the 2 F 1 's in Brafman's generating functions admit quadratic transformations, so that in essence, they are Legendre functions.…”

Section: Final Remarksmentioning

confidence: 99%

“…6 of Ref. 13.) Such 'ladder operators,' which increment and decrement ν and/or µ, come from the contiguity relations of 2 F 1 .…”

Section: Appendix Associated Legendre Functions In Closed Formmentioning

confidence: 99%

“…For this, define algebraic functions h ± , k ± trigonometrically byh ± (cosh ξ) = (sinh ξ) −1 ± cosh(ξ/3) + sinh ξ 3 sinh(ξ/3) 1/(A.6a) k ± (cos θ) = (sin θ) −1 cos(θ/3) ± sin θ 3 sin(θ/3) cosh ξ) = 3 (3/8)(1∓1) Γ(1 ∓ 1/4) −1 h ± (cosh ξ), cos θ) = 3 (3/8)(1∓1) Γ(1 ∓ 1/4) −1 k ± (cos θ). (A.7b) (The plus formulas were derived in Ref 13. and the minus formulas follow from them, the normalization factors coming from the condition (A.2).)…”

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Algebraic Generating Functions for Gegenbauer Polynomials

Maier¹

2018

Frontiers in Orthogonal Polynomials and q-Series

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It is shown that several of Brafman's generating functions for the Gegenbauer polynomials are algebraic functions of their arguments, if the Gegenbauer parameter differs from an integer by one-fourth or one-sixth. Two examples are given, which come from recently derived expressions for associated Legendre functions with octahedral or tetrahedral monodromy. It is also shown that if the Gegenbauer parameter is restricted as stated, the Poisson kernel for the Gegenbauer polynomials can be expressed in terms of complete elliptic integrals. An example is given.

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“…These appear as identities I 4 (i), I 4 (ii), and I 4 (ii) in [23] and are really quadratic hypergeometric transformations in disguise.…”

Section: Tetrahedral Formulas (Schwarz Class Iii)mentioning

confidence: 99%

Associated Legendre Functions and Spherical Harmonics of Fractional Degree and Order

Maier

2017

Constr Approx

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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.

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Legendre functions of fractional degree: transformations and evaluations

Cited by 7 publications

References 39 publications

Reversible reactions controlled by surface diffusion on a sphere

Reversible reactions controlled by surface diffusion on a sphere

Algebraic Generating Functions for Gegenbauer Polynomials

Associated Legendre Functions and Spherical Harmonics of Fractional Degree and Order

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