Jacobi Polynomials as Spherical Harmonics

Product Formula for Jacobi Polynomials, Spherical Harmonics and Generalized Bessel Function of Dihedral Type

Abstract. We work out the expression of the generalized Bessel function of B2-type derived in [4]. This is done using Dijskma and Koornwinder’s product formula for Jacobi polynomials and the obtained expression is given by multiple integrals involving only a normalized modified Bessel function and two symmetric Beta distributions. We think of that expression as the major step toward the explici...

Traceless Symmetric Tensor Approach to Legendre Polynomials and Spherical Harmonics

In these notes I will describe the separation of variable technique for solving Laplace’s equation, using spherical polar coordinates. The solutions will involve Legendre polynomials for cases with azimuthal symmetry, and more generally they will involve spherical harmonics. I will construct these solutions using traceless symmetric tensors, but in Lecture Notes 8 I describe how the solutions i...

Spherical Harmonics

5 Spherical Harmonics 7 5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.2.1 Harmonic expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.2.2 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

3 Generalized Spherical Harmonics

Optimal Spherical Harmonics Projection

We introduce the reader to the mathematics behind projection of n-dimensional vectors into a basis on n-dimensional space, where n can be anything upto and including infinity. We show how ideally one wants to project into the duals of a basis if this basis is not orthonormal, and provide the mathematics to formulate this operation in matrix form. The second part of the article discusses spheric...