Add hypergeometric function of a matrix argument

Overview

Provide an implementation for the hypergeometric function of a matrix argument, a generalization of the hypergeometric series.

Description

The function is commonplace in the densities of matrix variates, e.g. several distributions in the Matrix Variate Distributions text by Gupta and Nagar. The function is also described on Wikipedia.

Expected Output

Implement both the function and its logarithm as functions that can be called within a Stan program:

• real hypergeometric_matrix()
• real log_hypergeometric_matrix()

Additional Resources

• Efficient algorithm for the function: Koev, P., & Edelman, A. (2006). The efficient evaluation of the hypergeometric function of a matrix argument. Mathematics of Computation, 75(254), 833–846. https://doi.org/10.1090/S0025-5718-06-01824-2
  • With C code for MATLAB: https://math.mit.edu/~plamen/software/mhgref.html
• R implementation using C++: https://cran.r-project.org/package=HypergeoMat
  • GitHub source code: https://github.com/stla/HypergeoMat/blob/master/src/HypergeoMat.cpp

Current Version:

v4.4.0

Start by reviewing the requested hypergeometric_matrix() and log_hypergeometric_matrix() entry points alongside the Koev and Edelman algorithm and the linked HypergeoMat implementations. The issue does not identify Stan Math files, tests, argument signatures, or acceptance criteria beyond exposing both functions to Stan programs, so substantial design and project research are needed.