Two-dimensional orthogonal polynomials,their associated sets and the co-recursive sets

This paper deals with the general theory of sets of polynomials verifying a (d+1)-order recurrence. The cased=2 is specially carried out. First, we introduce the notion of associated set of a set of monic polynomials. A general formula for successive associated polynomials is given. The co-recursive sets of a two-dimensional orthogonal set are introduced. We calculate the corresponding formal Stieltjes functions.Finally, we determine the self-associated two-dimensional orthogonal sets and we show they are classical two-dimensional orthogonal sets, that is to say, their set of derivatives is also a two-dimensional orthogonal set.