1.3 Determinant Form for KummerU Function of a Matrix Argument

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

$$𝔘(a,b;𝒁)=\frac{1}{{\mathrm{\Gamma }}_p(a)}{\int }_{𝑿\succ 0}\text{etr}(-𝒁𝑿)det{(𝑿)}^{a-n}det{(𝑰+𝑿)}^{b-a-n}\text{d}𝑿,$$

where $𝒁,𝑿$ are $n\times n$ complex-valued symmetric positive-definite matrices, $\text{Re}(a)\ge n$, and ${\mathrm{\Gamma }}_p(a)$ is the multivariate Gamma function [15]. The definition of function $𝔘(a,b;𝒁)$ closely corresponds to the definition of the scalar KummerU function $U(a,b;z)$ given by

$$U(a,b;z)=\frac{1}{\mathrm{\Gamma }(a)}{\int }_0^{+\mathrm{\infty }}{e}^{-zx}{x}^{a-1}{(1+x)}^{b-a-1}\text{d}x.$$

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]:

$$\underset{c\to +\mathrm{\infty }}{lim}{}_2{}^{}𝔉_1^{}(a,b;c;𝑰-c{𝒁}^{-1})=det{(𝒁)}^b𝔘(b,b-a+n;𝒁),$$

where ${}_2{}^{}𝔉_1^{}$ 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

$$𝔘(a,b;𝒁)=\frac{1}{{\prod }_{1\le i\le j\le n}({\lambda }_i-{\lambda }_j)}det(𝛀),$$

where ${\lambda }_i,i=1,\mathrm{\dots },n$, are the non-repeating eigenvalues of $𝒁$, and $𝛀$ is an $n\times n$ matrix whose $(i,j)$-th element is given by

$${[𝛀]}_{ij}=U(a-j+1,a-b+1;{\lambda }_i),$$

and $U(a,b;z)$ is the scalar KummerU function as mentioned earlier.

While the above formula allows us to express $𝔘(a,b;𝒁)$ in an elegant way in terms of determinant of matrix of $U(a,b;z)$, computing $𝔘(a,b;𝒁)$ 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].