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 eigenvalues \(x_1\), \(\ldots\), \(x_n\) 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 Eldeman (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)

hypergeomPFQ(m = 15, a = c(3,4), b = c(5,6,7), x = cbind(c(0.1,0.4),c(0.4,0.1)))
#> [1] 1.011526

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 eigenvalues.