hypergeomatrix

Evaluation of the hypergeometric function of a matrix argument (Koev & Edelman's algorithm)

Let \((a_1, \ldots, a_p)\) and \((b_1, \ldots, b_q)\) be two vectors of real or complex numbers, possibly empty, \(\alpha > 0\) and \(X\) a real symmetric or a complex Hermitian matrix. The corresponding hypergeometric function of a matrix argument is defined by

\[{}_pF_q^{(\alpha)} \left(\begin{matrix} a_1, \ldots, a_p \\\\ b_1, \ldots, b_q\end{matrix}; X\right) = \sum_{k=0}^{\infty}\sum_{\kappa \vdash k} \frac{{(a_1)}_{\kappa}^{(\alpha)} \cdots {(a_p)}_{\kappa}^{(\alpha)}} {{(b_1)}_{\kappa}^{(\alpha)} \cdots {(b_q)}_{\kappa}^{(\alpha)}} \frac{C_{\kappa}^{(\alpha)}(X)}{k!}.\]

The inner sum is over the integer partitions \(\kappa\) of \(k\) (which we also denote by \(|\kappa| = k\)). The symbol \({(\cdot)}_{\kappa}^{(\alpha)}\) is the generalized Pochhammer symbol, defined by

\[{(c)}^{(\alpha)}_{\kappa} = \prod_{i=1}^{\ell}\prod_{j=1}^{\kappa_i} \left(c - \frac{i-1}{\alpha} + j-1\right)\]

when \(\kappa = (\kappa_1, \ldots, \kappa_\ell)\). Finally, \(C_{\kappa}^{(\alpha)}\) is a Jack function. Given an integer partition \(\kappa\) and \(\alpha > 0\), and a real symmetric or complex Hermitian matrix \(X\) of order \(n\), the Jack function

\[C_{\kappa}^{(\alpha)}(X) = C_{\kappa}^{(\alpha)}(x_1, \ldots, x_n)\]

is a symmetric homogeneous polynomial of degree \(|\kappa|\) in the eigen values \(x_1\), \(\ldots\), \(x_n\) of \(X\).

The series defining the hypergeometric function does not always converge. See the references for a discussion about the convergence.

The inner sum in the definition of the hypergeometric function is over all partitions \(\kappa \vdash k\) but actually \(C_{\kappa}^{(\alpha)}(X) = 0\) when \(\ell(\kappa)\), the number of non-zero entries of \(\kappa\), is strictly greater than \(n\).

For \(\alpha=1\), \(C_{\kappa}^{(\alpha)}\) is a Schur polynomial and it is a zonal polynomial for \(\alpha = 2\). In random matrix theory, the hypergeometric function appears for \(\alpha=2\) and \(\alpha\) is omitted from the notation, implicitely assumed to be \(2\).

Koev and Edelman (2006) provided an efficient algorithm for the evaluation of the truncated series

\[\sideset{_p^m}{_q^{(\alpha)}}F \left(\begin{matrix} a_1, \ldots, a_p \\\\ b_1, \ldots, b_q\end{matrix}; X\right) = \sum_{k=0}^{m}\sum_{\kappa \vdash k} \frac{{(a_1)}_{\kappa}^{(\alpha)} \cdots {(a_p)}_{\kappa}^{(\alpha)}} {{(b_1)}_{\kappa}^{(\alpha)} \cdots {(b_q)}_{\kappa}^{(\alpha)}} \frac{C_{\kappa}^{(\alpha)}(X)}{k!}.\]

Hereafter, \(m\) is called the truncation weight of the summation (because \(|\kappa|\) is called the weight of \(\kappa\)), the vector \((a_1, \ldots, a_p)\) is called the vector of upper parameters while the vector \((b_1, \ldots, b_q)\) is called the vector of lower parameters. The user has to supply the vector \((x_1, \ldots, x_n)\) of the eigenvalues of \(X\).

For example, to compute

\[\sideset{_2^{15}}{_3^{(2)}}F \left(\begin{matrix} 3, 4 \\\\ 5, 6, 7\end{matrix}; 0.1, 0.4\right)\]

you have to enter

hypergeomat 15 2 [3.0, 4.0], [5.0, 6.0, 7.0] [0.1, 0.4]

We said that the hypergeometric function is defined for a real symmetric matrix or a complex Hermitian matrix \(X\). Thus the eigenvalues of \(X\) are real. However we do not impose this restriction in hypergeomatrix. The user can enter any list of real or complex numbers for the eigenvalues.

Gaussian rational numbers

The library allows to use Gaussian rational numbers, i.e. complex numbers with a rational real part and a rational imaginary part. The Gaussian rational number \(a + ib\) is obtained with a +: b, e.g. (2%3) +: (5%2). The imaginary unit usually denoted by \(i\) is represented by e(4):

ghci> import Math.HypergeoMatrix
ghci> import Data.Ratio
ghci> alpha = 2%1
ghci> a = (2%7) +: (1%2)
ghci> b = (1%2) +: (0%1)
ghci> c = (2%1) +: (3%1)
ghci> x1 = (1%3) +: (1%4)
ghci> x2 = (1%5) +: (1%6)
ghci> hypergeomat 3 alpha [a, b] [c] [x1, x2]
26266543409/25159680000 + 155806638989/3698472960000*e(4)

Univariate case

For \(n = 1\), the hypergeometric function of a matrix argument is known as the generalized hypergeometric function. It does not depend on \(\alpha\). The case of \(\sideset{_{2\thinspace}^{}}{_1^{}}F\) is the most known, this is the Gauss hypergeometric function. Let's check a value. It is known that

\[\sideset{_{2\thinspace}^{}}{_1^{}}F \left(\begin{matrix} 1/4, 1/2 \\\\ 3/4\end{matrix}; 80/81\right) = 1.8.\]

Since \(80/81\) is close to \(1\), the convergence is slow. We compute the truncated series below for \(m = 300\).

ghci> h <- hypergeomat 300 2 [1/4, 1/2] [3/4] [80/81]
ghci> h
1.7990026528192298

References

• Plamen Koev and Alan Edelman. The efficient evaluation of the hypergeometric function of a matrix argument. Mathematics of computation, vol. 75, n. 254, 833-846, 2006.

• Robb Muirhead. Aspects of multivariate statistical theory. Wiley series in probability and mathematical statistics. Probability and mathematical statistics. John Wiley & Sons, New York, 1982.

• A. K. Gupta and D. K. Nagar. Matrix variate distributions. Chapman and Hall, 1999.