Minimal solutions of three-term recurrence relations and orthogonal polynomials

Gautschi, Walter

doi:10.1090/S0025-5718-1981-0606512-6

Minimal solutions of three-term recurrence relations and orthogonal polynomials
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Math. Comp. 36 (1981), 547-554 Request permission

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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