The Triangular Properties of Associated Legendre Functions Using The Vectorial Addition Theorem For Spherical Harmonics (original) (raw)
Derivatives of addition theorems for Legendre functions
The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 1995
Differentiation of the well-known addition theorem for Legendre polynomials produces results for sums over order m of products of various derivatives of associated Legendre functions. The same method is applied to the corresponding addition theorems for vector and tensor spherical harmonics. Results are also given for Chebyshev polynomials of the second kind, corresponding to 'spin-weighted' associated Legendre functions, as used in studies of distributions of rotations.
Associated Legendre Functions and Spherical Harmonics of Fractional Degree and Order
Constructive Approximation, 2017
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.
2020
The solution in hyperspherical coordinates for NNN dimensions is given for a general class of partial differential equations of mathematical physics including the Laplace, wave, heat and Helmholtz, Schrodinger, Klein-Gordon and telegraph equations and their combinations. The starting point is the Laplacian operator specified by the scale factors of hyperspherical coordinates. The general equation of mathematical physics is solved by separation of variables leading to the dependencies: (i) on time by the usual exponential function; (ii) on longitude by the usual sinusoidal function; (iii) on radius by Bessel functions of order generally distinct from cylindrical or spherical Bessel functions; (iv) on one latitude by associated Legendre functions; (v) on the remaining latitudes by an extension, namely the hyperspherical associated Legendre functions. The original associated Legendre functions are a particular case of the Gaussian hypergeometric functions, and the hyperspherical associ...
A triangular property of the associated Legendre functions
Journal of Mathematical Physics, 1990
The property of the associated Legendre functions with non-negative integer indices, Pmn(z), described by the formula: Pmn (cos β)=(−1)m(a/c)n ∑n−mk=0 (n+mk) (−b/a)n−k ×Pmn−k(cos γ), where a,b,c are the sides of an assigned triangle and α,β,γ the respective opposite angles, is introduced. A useful application of this series in simplifying the calculation of collisional electron–atom cross sections higher than the dipole is mentioned. A proof of the stated identity by use of the Gegenbauer polynomials and of their generating function is given.
On computation and use of Fourier coefficients for associated Legendre functions
Journal of Geodesy, 2016
The computation of spherical harmonic series in very high resolution is known to be delicate in terms of performance and numerical stability. A major problem is to keep results inside a numerical range of the used data type during calculations as under-/overflow arises. Extended data types are currently not desirable since the arithmetic complexity will grow exponentially with higher resolution levels. If the associated Legendre functions are computed in spectral domain then regular grid transformations can be applied highly efficiently and convenient for derived quantities as well. In this article we compare three recursive computations of the associated Legendre functions as trigonometric series, thereby ensuring a defined numerical range for each constituent wave-number, separately. The results to high degree and order show the numerical strength of the proposed method. First, the evaluation of Fourier coefficients of the associated Legendre functions has been done with respect to the floating-point precision requirements. Secondly, the numerical accuracy in the cases of standard Double and long Double precision arithmetic is demonstrated. Following Bessel's inequality the obtained accuracy estimates of the Fourier coefficients are directly transferable to the associated Legendre functions themselves and to derived functionals as well. Therefore, they can provide an essential insight to modern geodetic applications that depend on efficient spherical harmonic analysis and synthesis beyond [5 × 5] arcmin resolution.
Rocky Mountain Journal of Mathematics, 2010
We introduce a new class of polynomials {P n }, that we call polar Legendre polynomials, they appear as solutions of an inverse Gauss problem of equilibrium position of a field of forces with n + 1 unit masses. We study algebraic, differential and asymptotic properties of this class of polynomials, that are simultaneously orthogonal with respect to a differential operator and a discrete-continuous Sobolev type inner product.
On Two Useful Identities in the Theory of Ellipsoidal Harmonics
Studies in Applied Mathematics, 2009
Two identities on ellipsoidal harmonics, which appear naturally in the theory of boundary value problems, are stated and proved. The first involves the ellipsoidal analogue of the Beltrami operator in spherical coordinates (also known as surface Laplacian). The second identity includes the tangential surface gradient operator defined as the cross product of the unit normal with the gradient operator on an ellipsoidal surface. In both cases, the basic spectral properties of these two operators, as they act on the surface ellipsoidal harmonics, are provided.
Some integral identities for spherical harmonics in an arbitrary dimension
Journal of Mathematical Chemistry, 2011
Spherical harmonics in an arbitrary dimension are employed widely in quantum theory. Spherical harmonics are also called hyperspherical harmonics when the dimension is larger than 3. In this paper, we derive some integral identities involving spherical harmonics in an arbitrary dimension.