You could start by trying to compile and work with the detailed sample code provided in the link on the GSoC project page.
Just to start the discussion: Does everyone agree that a variety of methods need to employed for calculating the functions for different ranges of parameters?
Absolutely. We need various methods for various parameter ranges and also for different precision ranges.
Taylor series methods for smaller values of |a| and |z|.
Yes, whereby it can be challenging deciding on the range of convergence.
Any other ideas?
I like your idea of Gauss-Jacobi quadrature methods. There is also a *grand new* example featuring generation of Gauss-Laguerre coefficients for multiple-precision. https://github.com/boostorg/multiprecision/blob/develop/example/gauss_laguer... In this project, we will be investigating a variety of calculational methods including Chebyshev polynomial expansion, rational approximation, Pade approximation, and perhaps others. There are also some cool expansions for some hypergeometric functions in series of Bessel functions. Best regards, Chris.