Support for Gegenbauer polynomials / hyperspherical harmonics?

[Dashiell Stander]

Hi! I'm working on problems with data defined on the n-sphere (n >= 128 or so). From looking around the documentation it seems like this library doesn't support that--do I have the right? Are you familiar with any other libraries that support it?

[Mikael Slevinsky]

Hi @dashstander, it only supports spherical harmonics on the 2-sphere.

I'm not sure full hyperspherical harmonic expansions on the 128-sphere are computationally tractable. But if you are looking for just the ultraspherical polynomials themselves, then yes, there are some routines for them here

FastTransforms/src/fasttransforms.h

Lines 72 to 80 in 0df512b

 /*! 
   \brief Pre-compute a factorization of the connection coefficients between ultraspherical polynomials in double precision so that ft_bfmv converts between expansions: 
   \f[ 
   \sum_{\ell=0}^{n-1} c_\ell^{(1)} C_\ell^{(\lambda)}(x) = \sum_{\ell=0}^{n-1} c_\ell^{(2)} C_\ell^{(\mu)}(x). 
   \f] 
   `norm1` and `norm2` govern the normalizations, either standard ( == 0) or orthonormalized ( == 1).\n 
   See also \ref ft_plan_ultraspherical_to_ultrasphericalf, \ref ft_plan_ultraspherical_to_ultrasphericall, and \ref ft_mpfr_plan_ultraspherical_to_ultraspherical. 
 */ 
 ft_tb_eigen_FMM * ft_plan_ultraspherical_to_ultraspherical(const int norm1, const int norm2, const int n, const double lambda, const double mu);