Integral expression for the stationary distribution of reflected Brownian motion in a wedge

For Brownian motion in a (two-dimensional) wedge with negative drift and oblique reflection on the axes, we derive an explicit formula for the Laplace transform of its stationary distribution (when it exists), in terms of Cauchy integrals and generalized Chebyshev polynomials. To that purpose, we solve a Carlemantype boundary value problem on a hyperbola, satisfied by the Laplace transforms of the boundary stationary distribution.