  We prove an operator level limit for the circular Jacobi $\beta$-ensemble. As
a result, we characterize the counting function of the limit point process via
coupled systems of stochastic differential equations. We also show that the
normalized characteristic polynomials converge to a random analytic function,
which we characterize via the joint distribution of its Taylor coefficients at
zero and as the solution of a stochastic differential equation system. We also
provide analogous results for the real orthogonal $\beta$-ensemble.
