Algorithms for classical orthogonal polynomials
Abstract
In this article explicit formulas for the recurrence equation pn+1(x) = (An x + Bn) pn(x) - Cn pn-1(x) and the derivative rules sigma(x) p'n(x) = alphan pn+1(x) + betan pn(x) + gamman pn-1(x) and sigma(x) p'n(x) = (alphan-tilde x + betan-tilde) pn(x) + gamman-tilde pn-1(x) respectively which are valid for the orthogonal polynomial solutions pn(x) of the differential equation sigma(x) y''(x) + r(x) y'(x) + lambdan y(x) = 0 of hypergeometric type are developed that depend only on the coefficients sigma(x) and tau(x) which themselves are polynomials w.r.t. x of degree not larger than 2 and 1, respectively. Partial solutions of this problem had beed previously published by Tricomi, and recently by Y\'a\~nez, Dehesa and Nikiforov.
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