Classical special functions of matrix arguments

This article focuses on a few of the most commonly used special functions and their key properties and defines an analytical approach to building their matrix-variate counterparts. To achieve this, we refrain from using any numerical approximation algorithms and instead rely on properties of matrices, the matrix exponential, and the Jordan normal form for matrix representation. We focus on the following functions: the Gamma function as an example of a univariate function with a large number of properties and applications; the Beta function to highlight the similarities and differences from adding a second variable to a matrix-variate function; and the Jacobi Theta function. We construct explicit function views and prove a few key properties for these functions. In the comparison section, we highlight and contrast other approaches that have been used in the past to tackle this problem.