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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1981-0606512-6
PII: S 0025-5718(1981)0606512-6
Keywords: Minimal solutions of three-term recurrence relations, orthogonal polynomials, moment problem, modified moments
Article copyright: © Copyright 1981 American Mathematical Society