Gibbs Sampling, Exponential Families and Orthogonal Polynomials
  We give families of examples where sharp rates of convergence to stationarity of the widely used Gibbs sampler are available. The examples involve standard exponential families and their conjugate priors. In each case, the transition operator is explicitly diagonalizable with classical orthogonal polynomials as eigenfunctions. 
Persi Diaconis, Kshitij Khare, Laurent Saloff-Coste