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Classification of classical orthogonal polynomials. (English) Zbl 0898.33002

The authors consider the differential equation \[ l_2(x)y''+ l_1(x)y'= \lambda_n y(x), \tag{\(*\)} \] , where \(x\in R\), \(l_1(x)= l_{11}x+ l_{10}\) and \(l_2(x)= l_{22}x^2+ l_{21}x+ l_{20}\) with certain coefficients \(l_{ij}\) while \(\lambda_n= n(n-1)l_{22}+ nl_{11}\) \((n\geq 0)\), and discuss different cases for which \((*)\) has polynomial solutions which satisfy certain orthogonality conditions. According to the value of \(l_{22}\) as well as to the signs of \(l_{21}^2- 4l_{22}l_{20}\) and \(l_{11}\) they distinguish six different cases, namely: The Jacobi, twisted Jacobi, Bessel, Laguerre, Hermite and twisted Hermite cases. For cach case they also derive related orthogonality relation.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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