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Let $\mathbf{W}_1$ and $\mathbf{W}_2$ be independent $n\times n$ complexcentral Wishart matrices with $m_1$ and $m_2$ degrees of freedom respectively.This paper is concerned with the extreme eigenvalue distributions ofdouble-Wishart matrices $(\mathbf{W}_1+\mathbf{W}_2)^{-1}\mathbf{W}_1$, whichare analogous to those of F matrices ${\bf W}_1 {\bf W}_2^{-1}$ and those ofthe Jacobi unitary ensemble (JUE). Defining $\alpha_1=m_1-n$ and$\alpha_2=m_2-n$, we derive new exact distribution formulas in terms of$(\alpha_1+\alpha_2)$-dimensional matrix determinants, with elements involvingderivatives of Legendre polynomials. This provides a convenient exactrepresentation, while facilitating a direct large-$n$ analysis with $\alpha_1$and $\alpha_2$ fixed (i.e., under the so-called "hard-edge" scaling limit); theanalysis is based on new asymptotic properties of Legendre polynomials andtheir relation with Bessel functions that are here established. Specifically,we present limiting formulas for the smallest and largest eigenvaluedistributions as $n \to \infty$ in terms of $\alpha_1$- and$\alpha_2$-dimensional determinants respectively, which agrees withexpectations from known universality results involving the JUE and the Laguerreunitary ensemble (LUE). We also derive finite-$n$ corrections for theasymptotic extreme eigenvalue distributions under hard-edge scaling, giving newinsights on universality by comparing with corresponding correction termsderived recently for the LUE. Our derivations are based on elementary algebraicmanipulations, differing from existing results on double-Wishart and relatedmodels which often involve Fredholm determinants, Painlev\'e differentialequations, or hypergeometric functions of matrix arguments.

Extreme Eigenvalue Distributions of Jacobi Ensembles: New Exact Representations, Asymptotics and Finite Size Corrections