HypergeoMat.jl documentation

Hypergeometric function of a matrix argument

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

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

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 $2$. This is the default value of $\alpha$ in the HypergeoMat package.

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

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

In the HypergeoMat package, $m$ 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 can enter either the matrix $X$ or the vector $(x_1, \ldots, x_n)$ of the eigenvalues of $X$.

For example, to compute

\[{{}_{2\!\!\!\!\!}}^{15}\! F_3^{(2)} \left(\begin{matrix} 3, 4 \\ 5, 6, 7\end{matrix}; \begin{pmatrix} 0.1 && 0.4 \\ 0.4 && 0.1 \end{pmatrix}\right)\]

you have to enter (recall that $\alpha=2$ is the default value)

hypergeomPQ(15, [3.0;4.0], [5.0;6.0;7.0], [0.1 0.4; 0.4 0.1])

We said that the hypergeometric function is defined for a real symmetric matrix or a complex Hermitian matrix $X$. However we do not impose this restriction in the HypergeoMat package. The user can enter any real or complex square matrix, or a real or complex vector of eigen values.

References

• Plamen Koev and Alan Edelman. The Efficient Evaluation of the Hypergeometric Function of a Matrix Argument. Mathematics of Computation, 75, 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.

Member functions

hypergeomPQ(m, a, b, x[, alpha])

hypergeomPQ(m, a, b, X[, alpha])

hypergeomPQ(m, a, b, x)

lmvgamma(z, p)

mvgamma(z, p)

BesselA(m, X, nu)

BesselA(m, x, nu)