Summary
Some properties of orthogonal (and generalized orthogonal) polynomial sets in two variables are obtained, in particular a characterization of such sets based on generating functions. Then those linear homogeneous partial differential eqnations of the form L[w]+λw=0, having a set of polynomials as solution, are characterized; and a detailed study is made of all such equations of second order whose polynomial solutions form an orthogonal (or generalized orthogonal) set.
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P. Appell -J. K. de Fériet,Fonctions Hypergéometriques et Hypersphériques; Polyno mes d' Hermite, Paris, Gauthier-Villars et Cie. (1926).
R. P. Boas Jr.,The Stieltjes moment problem for functions of bounded variation, Bull. Amer. Math. Soc. 45, 339–404 (1939).
F. Didon,Étude de certaines function analogues aux fonctions X n de Legendre, Ann. Sci. Ecole Normale sup. 1st ser. v. 5 (1868).
C. Hermite,Oeuvres, v. 2 (1908), 309–346, Paris, Gauthier-Villars et Cie.
D. Jackson,Formal properties of orthogonal polynomials in two variables, Duke Math. Journal, v. 2 (1936), 423–434.
H. L. Krall -I. M. Shefeer,A characterization of orthogonal polynomials, Journal of Math. Analysis and Applications, v. 8 (1964), 232–244.
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Supported byN.S.F. Grant GP-5311.
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Krall, H.L., Sheffer, I.M. Orthogonal polynomials in two variables. Annali di Matematica 76, 325–376 (1967). https://doi.org/10.1007/BF02412238
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DOI: https://doi.org/10.1007/BF02412238