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  "body": "\n## Summary\n\nA hypergeometric function of a matrix argument ${}_pF_q^{(\\alpha)}(a; b; X)$ is a generalisation of the [[generalized hypergeometric function|VDBAW9PN88]] in which the variable is a matrix $X$.\nThe series runs over [[integer partitions|0JV30XN73Z]].\nThe powers $z^k$ are replaced by Jack polynomials in the eigenvalues of $X$, and the rising factorials by generalized Pochhammer symbols.\nThese functions appear in multivariate statistics and random matrix theory, where densities of eigenvalues of Wishart and related matrices are written with them.\n\n## Explanations\n\n### L1 human\n\nFor a number $z$, the series ${}_pF_q$ is a sum over $k$ of rational coefficients times $z^k / k!$.\nFor a matrix, the role of $z^k$ is played by a family of symmetric polynomials of degree $k$ in the eigenvalues, one for each way of splitting $k$ into parts.\nThese polynomials add up to $(\\operatorname{tr} X)^k$, so the matrix series keeps track of finer information than the trace alone.\nThe simplest case is ${}_0F_0(X) = e^{\\operatorname{tr} X}$, which comes out because the polynomials of degree $k$ sum to $(\\operatorname{tr} X)^k$.\nThe parameter $\\alpha$ selects the setting: $\\alpha = 2$ for real symmetric matrices and $\\alpha = 1$ for complex Hermitian matrices.\n",
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