Gap Probability Distribution of the Jacobi Unitary Ensemble: An Elementary Treatment, from Finite $n$ to Double Scaling

In this paper, we study the gap probability problem of the (symmetric) Jacobi unitary ensemble of Hermitian random matrices, namely the probability that the interval $(-a,a)(0<a<1)$ is free of eigenvalues. Using the ladder operator technique for orthogonal polynomials and the associated supplementary conditions, we derive three quantities instrumental in the gap probability, denoted by $H_n(a)$, $R_n(a)$ and $r_n(a)$. We find that each one satisfies a second order differential equation. We show that after a double scaling, the large second order differential equation in the variable $a$ with $n$ as parameter satisfied by $H_n(a)$, can be reduced to the Jimbo-Miwa-Okamoto $\sigma$ form of the Painlev'{e} V equation.