Software to compute spherical harmonics in higher than 3 dimensions (100 or maybe 500 dimensions)? I have been trying to find an implementation of Spherical harmonics for higher dimensional data but I couldnt find anything in Sage, Mathematica, Matlab. Does anyone have any idea of a standard/fast implementation ?
MORE DETAILS: I am adding more information about the expression that I am interested in. This comes from equation 13 from this paper on the eigenfunctions of dot product kernels.

Moreover, the Legendre Polynomials may be expanded into an orthonormal basis
  of spherical harmonics $Y^d_{n,j}$ by the Funk-Hecke equation to obtain: 
  $$P^n_d(x.y) = \frac{|S_{d-1}|}{N(d,n)} \sum_{j=1}^{N(d,n)} Y_{n,j}^d(x) Y_{n,j}^d(y)$$

Here $x,y$ are vectors in $R^d$ with unit norm and I am interested in computing the spherical harmonic $Y_{n,j}^d$ at the points $x$ and $y$. 
 A: Does this work for you?

The HFT.m Mathematica software package performs symbolic manipulation of expressions that arise in the study of harmonic functions. This software, which is available electronically without charge, can perform symbolic calculations that would take a prohibitive amount of time if done without a computer. For example, the Poisson integral of any polynomial can be computed exactly.
Some of the capabilities of this software:

*

*symbolic calculus in $\mathbb{R}^n$

*Dirichlet problem for balls, quadratic regions, annular regions, and exteriors of balls in $\mathbb{R}^n$

*Neumann problem for balls and exteriors of balls in $\mathbb{R}^n$

*biDirichlet problem for balls in $\mathbb{R}^n$

*the Bergman projection problem for balls in $\mathbb{R}^n$

*$\color{red}{\text{finding bases for spherical harmonics in }\mathbb{R}^n}$

*explicit formulas for zonal harmonics in $\mathbb{R}^n$

*manipulations with the Kelvin transform

*Schwarz functions for balls in $\mathbb{R}^n$

*harmonic conjugation in $\mathbb{R}^2$

