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The probabilities for gaps in the eigenvalue spectrum of the finite dimension$ N \times N $ random matrix Hermite and Jacobi unitary ensembles on somesingle and disconnected double intervals are found. These are cases where areflection symmetry exists and the probability factors into two other relatedprobabilities, defined on single intervals. Our investigation uses the systemof partial differential equations arising from the Fredholm determinantexpression for the gap probability and the differential-recurrence equationssatisfied by Hermite and Jacobi orthogonal polynomials. In our study we findsecond and third order nonlinear ordinary differential equations defining theprobabilities in the general $N$ case. For N=1 and N=2 the probabilities andthus the solution of the equations are given explicitly. An asymptoticexpansion for large gap size is obtained from the equation in the Hermite case,and also studied is the scaling at the edge of the Hermite spectrum as $ N \to\infty $, and the Jacobi to Hermite limit; these last two studies makecorrespondence to other cases reported here or known previously. Moreover, thedifferential equation arising in the Hermite ensemble is solved in terms of anexplicit rational function of a {Painlev\'e-V} transcendent and its derivative,and an analogous solution is provided in the two Jacobi cases but this timeinvolving a {Painlev\'e-VI} transcendent.Comment: 32 pages, Latex2

Gap probabilities for edge intervals in finite Gaussian and Jacobi unitary matrix ensembles