ON 2-DIMENSIONAL DEFINITE ORTHOGONAL SYSTEMS AND A LOWER-BOUND FOR THE NUMBER OF NODES OF ASSOCIATED CUBATURE FORMULAS

In a comprehensive investigation in the 1960s Krall and Sheffer [Ann. Mat. Pura Appl., 76 (1967), pp. 325-376] characterized all bivariate o rthogonal polynomial systems which are generated by a second-order dif ferential equation. Actually, they prove that these nine systems are w eakly orthogonal and (positive) definite except possibly for two syste ms. Their paper is completed by showing that these systems are also de finite and by determining all parameters for which the classical posit ive definite systems remain definite. The authors further derive an ex plicit form of the three-term recursion formulae for all systems. In a ddition, it is shown that for the associated cubature problem Moller's lower bound applies.