Classical discrete symplectic ensembles on the linear and exponential lattice: skew orthogonal polynomials and correlation functions

The eigenvalue probability density function for symplectic invariant random matrix ensembles can be generalised to discrete settings involving either a linear or exponential lattice. The corresponding correlation functions can be expressed in terms of certain discrete, and $q$, skew orthogonal polynomials respectively. We give a theory of both of these classes of polynomials, and the correlation kernels determining the correlation functions, in the cases that the weights for the corresponding discrete unitary ensembles are classical. Crucial for this are certain difference operators which relate the relevant symmetric inner products to the skew symmetric ones, and have a tridiagonal action on the corresponding (discrete or $q$) orthogonal polynomials.