Classification of classical orthogonal polynomials
Classification of classical orthogonal polynomials
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.
- Korea Advanced Institute of Science and Technology Korea (Republic of)
Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), orthogonality, classical orthogonal polynomials
Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), orthogonality, classical orthogonal polynomials
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