The Laguerre, Charlier, and Meixner polynomials are polynomials in two variables: in x they are classical orthogonal polynomials, and in a parameter b they are type \(R_I\) orthogonal polynomials. Thus they have two types of orthogonality relations. Remarkably, their type \(R_I\) moments are identified as the orthogonal polynomial moments for another set of classical polynomials. A general notion of moment duality is introduced for polynomials in two variables. This program is continued for two and three parameter Askey-Wilson polynomials. The equality of the moments is equivalent to the equality of two continued fractions.
