Large-Parameter Asymptotic Expansions for the Legendre and Allied Functions
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Associated Legendre Functions and Spherical Harmonics of Fractional Degree and Order
Constructive Approximation manuscript No. (will be inserted by the editor) Associated Legendre Functions and Spherical Harmonics of Fractional Degree and Order Robert S. Maier Received: date / Accepted: date Abstract Trigonometric formulas are derived for certain families of associated Legendre functions of fractional degree and order, for use in approximation the- ory. These functions are algebraic, and when viewed as Gauss hypergeometric functions, belong to types classified by Schwarz, with dihedral, tetrahedral, or oc- tahedral 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 (gener- alized) 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. Keywords Associated Legendre function Algebraic function Spherical · · harmonic Solid harmonic Jacobi polynomial Heun polynomial Ladder · · · · operator Mathematics Subject Classification (2000) 33C45 33C47 33C55 22E70 · · · 1 Introduction µ µ The first-kind associated Legendre functions Pν (z), or the Ferrers versions Pν (z), µ µ are classical. (Pν (z) and Pν (z) are continuations of each other, with respective real domains z (1, ) and z ( 1, 1).) The roles they play when the degree ν ∈ ∞ ∈ − Robert S. -
On the Derivative of the Associated Legendre Function
J Math Chem (2011) 49:1436–1477 DOI 10.1007/s10910-011-9826-3 ORIGINAL PAPER On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree (with applications to the construction of the associated Legendre function of the second kind of integer degree and order) Radosław Szmytkowski Received: 13 October 2010 / Accepted: 18 April 2011 / Published online: 13 May 2011 © The Author(s) 2011. This article is published with open access at Springerlink.com Abstract In our recent works (R. Szmytkowski, J Phys A 39:15147, 2006; corrigendum: 40:7819, 2007; addendum: 40:14887, 2007), we have investigated the derivative of the Legendre function of the first kind, Pν(z), with respect to its degree ν. In the present work, we extend these studies and construct several representations ±m of the derivative of the associated Legendre function of the first kind, Pν (z), with respect to the degree ν,form ∈ N. At first, we establish several contour-integral ±m representations of ∂ Pν (z)/∂ν. They are then used to derive Rodrigues-type for- ±m mulas for [∂ Pν (z)/∂ν]ν=n with n ∈ N. Next, some closed-form expressions for ±m [∂ Pν (z)/∂ν]ν=n are obtained. These results are applied to find several representa- tions, both explicit and of the Rodrigues type, for the associated Legendre function of ±m( ) the second kind of integer degree and order, Qn z ; the explicit representations are suitable for use for numerical purposes in various regions of the complex z-plane. Fi- 2 m 2 m m nally, the derivatives [∂ Pν (z)/∂ν ]ν=n, [∂ Qν (z)/∂ν]ν=n and [∂ Qν (z)/∂ν]ν=−n−1, −m all with m > n, are evaluated in terms of [∂ Pν (±z)/∂ν]ν=n. -
Hypergeometric Type Functions and Their Symmetries
Ann. Henri Poincar´e Online First c 2013 The Author(s). This article is published with open access at Springerlink.com Annales Henri Poincar´e DOI 10.1007/s00023-013-0282-4 Hypergeometric Type Functions and Their Symmetries Jan Derezi´nski Abstract. The paper is devoted to a systematic and unified discussion of various classes of hypergeometric type equations: the hypergeometric equation, the confluent equation, the F1 equation (equivalent to the Bessel equation), the Gegenbauer equation and the Hermite equation. In partic- ular, recurrence relations of their solutions, their integral representations and discrete symmetries are discussed. Contents 1. Introduction 1.1. Classification 1.2. Properties of Hypergeometric Type Operators 1.3. Hypergeometric Type Functions 1.4. Degenerate Case 1.5. Canonical Forms 1.6. Hypergeometric Type Polynomials 1.7. Parametrization 1.8. Group-Theoretical Background 1.9. Comparison with the Literature 2. Preliminaries 2.1. Differential Equations 2.2. The Principal Branch of the Logarithm and the Power Function 2.3. Contours 2.4. Reflection Invariant Differential Equations 2.5. Regular Singular Points 2.6. The Gamma Function 2.7. The Pochhammer Symbol 3. The 2F1 or the Hypergeometric Equation 3.1. Introduction J. Derezi´nski Ann. Henri Poincar´e 3.2. Integral Representations 3.3. Symmetries 3.4. Factorization and Commutation Relations 3.5. Canonical Forms 3.6. The Hypergeometric Function 3.7. Standard Solutions: Kummer’s Table 3.7.1. Solution ∼ 1at0 3.7.2. Solution ∼ z−α at 0 3.7.3. Solution ∼ 1at1 3.7.4. Solution ∼ (1 − z)−β at 1 3.7.5. Solution ∼ z−a at ∞ 3.7.6. -
Lecture 6 Notes, Electromagnetic Theory I Dr
Lecture 6 Notes, Electromagnetic Theory I Dr. Christopher S. Baird University of Massachusetts Lowell 1. Associated Legendre Polynomials - We now return to solving the Laplace equation in spherical coordinates when there is no azimuthal symmetry by solving the full Legendre equation for m = 0 and m ≠ 0: m 2 d 2 d Pl x m m [1− x ][l l1− ]P x=0 where x=cos dx dx 1−x2 l - Once this equation is solved, the general solution for the Laplace equation in spherical coordinates will have the form: =∑ l −l−1 m=0 r , , Al r B l r Am0 Bm0 P l cos l ∑ l −l −1 i m −i m m Al r Bl r Am e B m e P l cos m≠0,l - Encouraged by the Rodrigues' form of the solution to the m = 0 Legendre equation, we try a m( )=( − 2)m/ 2 d m ( ) solution of the form P l x 1 x dx m g x and substitute in: m1 m m =− − 2m/2[ d ] 2 d 2 − 2m/2−1− d − 2m/ 2 0 m x 1 x g x m g x x 1 x m g x 1 x dxm 1 dxm dx m m2 m1 − 2m/21[ d ]− − 2m/ 2 d 1 x g x m 2 x 1 x g x dxm 2 dxm 1 m m 2 m/2 d 2 2 m/ 2−1 d [l l1]1−x g x−m 1−x g x dxm dx m - Collect all similar terms: m 2 m1 m = − 2[ d d ]− [ ] d [ − ] d 0 1 x g x 2 m 1 x g x l l 1 m m 1 g x dxm dx2 dxm 1 dxm - We want to move as much as possible inside the derivatives. -
Associated Legendre Functions and Spherical Harmonics of Fractional Degree and Order
Constr Approx (2018) 48:235–281 https://doi.org/10.1007/s00365-017-9403-5 Associated Legendre Functions and Spherical Harmonics of Fractional Degree and Order Robert S. Maier1 Received: 25 February 2017 / Accepted: 30 August 2017 / Published online: 13 November 2017 © The Author(s) 2017. This article is an open access publication Abstract Trigonometric formulas are derived for certain families of associated Leg- endre 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 hyper- geometric 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 repre- sentations 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. Keywords Associated Legendre function · Algebraic function · Spherical harmonic · Solid harmonic · Jacobi polynomial · Heun polynomial · Ladder operator Mathematics Subject Classification 33C45 · 33C47 · 33C55 · 22E70 Communicated by Tom H. Koornwinder. B Robert S. Maier [email protected] 1 Departments of Mathematics and Physics, University of Arizona, Tucson, AZ 85721, USA 123 236 Constr Approx (2018) 48:235–281 1 Introduction μ μ The first-kind associated Legendre functions Pν (z), or the Ferrers versions Pν (z), μ μ are classical.