KummerU function is the confluent hypergeometric function of the second kind. In this post, a straightforward method of expressing the KummerU function of a matrix argument in terms of the determinant of a matrix of scalar KummerU functions is presented.
KummerU function of a matrix argument [8, Definition 1.3.6] given by
where are complex-valued symmetric positive-definite matrices, , and is the multivariate Gamma function [15]. The definition of function closely corresponds to the definition of the scalar KummerU function given by
We begin by noting that the determinant form for generalized hypergeometric functions are known due to [16, Eqn. 34]. Hence, KummerU function of a matrix argument may be expressed in its determinant form in a straightforward manner using the relation [8, Definition 1.3.6]:
where is the Gaussian hypergeometric function of a matrix argument. A corresponding relation for scalar KummerU function is given in KummerU function.
Using the relation above and [16, Eqn. 34], the determinant form for KummerU function of a matrix argument can be simplified to
where , are the non-repeating eigenvalues of , and is an matrix whose -th element is given by
and is the scalar KummerU function as mentioned earlier.
While the above formula allows us to express in an elegant way in terms of determinant of matrix of , computing in terms of Zonal polynomials may be more efficient. For an example of using Zonal polynomials to evaluate generalized hypergeometric functions in MATLAB, see [10].
First published: 14th Dec. 2018 on aravindhk-math.blogspot.com
Modified: 16th Dec. 2023 – Style updates for LaTeX