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Mathematics of Computation

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Minimal solutions of three-term recurrence relations and orthogonal polynomials

Author: Walter Gautschi
Journal: Math. Comp. 36 (1981), 547-554
MSC: Primary 33A65; Secondary 65D20, 65D30
MathSciNet review: 606512
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Abstract: We observe that the well-known recurrence relation $ {p_{n + 1}}(z) = (z - {a_n}){p_n}(z) - {b_n}{p_{n - 1}}(z)$ for orthogonal polynomials admits a "minimal solution" if z is outside the spectrum of the mass distribution $ ds(t)$ with respect to which the polynomials are orthogonal and if the moment problem for this distribution is determined. The minimal solution, indeed is $ {f_n}(z) = \smallint {p_n}(t)\,ds(t)/(z - t)$, and can be computed accurately by means of the author's continued fraction algorithm. An application is made to special Gauss-type quadrature formulas.

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Keywords: Minimal solutions of three-term recurrence relations, orthogonal polynomials, moment problem, modified moments
Article copyright: © Copyright 1981 American Mathematical Society