pyhypergeomatrix

Hypergeometric functions of a matrix argument.

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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

>>> from pyhypergeomatrix.hypergeomat import hypergeomPQ
>>> hypergeomPQ(15, [3, 4], [5, 6, 7], [0.1, 0.4], 2)

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 pyhypergeomatrix. The user can enter any list of real or complex numbers for the eigenvalues.

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 $m = 300$.

>>> from pyhypergeomatrix.hypergeomat import hypergeomPQ
>>> hypergeomPQ(300, [1/4, 1/2], [3/4], [80/81])
1.79900265281923

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.