Some properties of orthogon~l (and generalized orthogonal) polynomial sets in two variables are obtained, in particular a characterization of such sets based on generating functions. Then those li+~ear homogeneous partial differential eqnations o[ the form Lily]-bkvo:O, having a set of polynomials as soluhon, are characterized; and a detailed study is made of all such equations of second order ~vhose polynomial solutions form an orthogonal (or generalized orthogonal) set. 1. -Introduction.

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Differential equations of infinite order of the form ~ M~(x)y~k)(~c)=),y(~c), qvhere Mk is a polynomial of degree <= k, have polynomial solutionv for suitable values of ~. Necessary and sufficient conditions are given in order that these solutions form a set of orthogonal polynomials. The cases ~vhere max t degree Mk I = 1 and 2 are studied in detail, k O. -I n t r o d u c t i o n .