An Abundance of Orthogonal Polynomials
- 1 March 1982
- journal article
- Published by Oxford University Press (OUP) in IMA Journal of Applied Mathematics
- Vol. 28 (2) , 161-167
- https://doi.org/10.1093/imamat/28.2.161
Abstract
For positive integers k, m the ordinary differential equations (1 − xm)x − (k − 1+ δxm + n(n + δ − 1)xm − 1P = 0. x −(k − 1 + mxm) + mnxm − 1 P = ), with k < m, δ > 1 − k, have orthogonal polynomial solutions with the degrees n restricted to the values n = 0, k, m, m+k, 2m, 2m+k,…. At m = 1 it is not necessary for k to be a positive integer and the solutions to the two classes of equations are Jacobi and Laguerre polynomials. At m = 2 the only permissible value for k is k = 1, and the solutions are Gegenbauer and Hermite polynomials. For larger m, new polynomials are obtained whenever m, k have no common denominator.Keywords
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