Painlev\'e expressions for LOE, LSE and interpolating ensembles

We consider an ensemble which interpolates the Laguerre orthogonal ensemble and the Laguerre symplectic ensemble. This interpolating ensemble was introduced earlier by the author and Rains in connection with a last passage percolation model with a symmetry condition. In this paper, we obtain a Painelev\'e V expression for the distribution of the rightmost particle of the interpolating ensemble. Special cases of this result yield the Painlev\'e V expressions for the largest eigenvalues of Laguerre orthogonal ensemble and Laguerre symplectic ensemble of finite size.