  In this paper, a fourth-order partial divided-difference equation on
quadratic lattices with polynomial coefficients satisfied by bivariate Racah
polynomials is presented. From this equation we obtain explicitly the matrix
coefficients appearing in the three-term recurrence relations satisfied by any
bivariate orthogonal polynomial solution of the equation. In particular, we
provide explicit expressions for the matrices in the three-term recurrence
relations satisfied by the bivariate Racah polynomials introduced by Tratnik.
Moreover, we present the family of monic bivariate Racah polynomials defined
from the three-term recurrence relations they satisfy, and we solve the
connection problem between two different families of bivariate Racah
polynomials. These results are then applied to other families of bivariate
orthogonal polynomials, namely the bivariate Wilson, continuous dual Hahn and
continuous Hahn, the latter two through limiting processes. The fourth-order
partial divided-difference equations on quadratic lattices are shown to be of
hypergeometric type in the sense that the divided-difference derivatives of
solutions are themselves solution of the same type of divided-difference
equations.
