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Partial differential equations having orthogonal polynomial solutions. (English) Zbl 0930.35032

Orthogonal polynomial solutions in two variables satisfying the second order partial differential equation with polynomial coefficients \[ L[u]\equiv Au_{xx}+2Bu_{xy}+Cu_{yy}+Du_{x}+Eu_{y}=\lambda u\tag{*} \] are considered [see H. L. Krall and I. M. Sheffer, Ann. Mat. Pura Appl., IV. Ser. 76, 325-376 (1967; Zbl 0186.38602)]. It is proved that if the differential equation (*) has an orthogonal polynomial system as solutions, then \(B^2\not=AC\) in any nonempty open subset of \(\mathbb R^2\) (nonparabolic case of (*)). In addition it is shown that if the equation (*) has an orthogonal polynomial system as solutions, then the differential operator \(L\) must be symmetrizable [see L. L. Littlejohn, Proc. Int. Symp., Segovia/Spain 1986, Lect. Notes Math. 1329, 98-124 (1988; Zbl 0653.42022)]. Introducing a weak orthogonal polynomial system of solutions to (*) the Rodrigues type formula was obtained too. To illustrate the results obtained some examples are given.

MSC:

35C05 Solutions to PDEs in closed form
33C47 Other special orthogonal polynomials and functions
33E30 Other functions coming from differential, difference and integral equations
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