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Special classes of orthogonal polynomials and corresponding quadratures of Gaussian type. (English) Zbl 1272.33013

Summary: In the first part of this survey paper, we present a short account on some important properties of orthogonal polynomials on the real line, including computational methods for constructing coefficients in the fundamental three-term recurrence relation for orthogonal polynomials, and mention some basic facts on Gaussian quadrature rules.
In the second part, we discuss our Mathematica package OrthogonalPolynomials (see the authors [Facta Univ., Ser. Math. Inf. 19, 17–36 (2004; Zbl 1081.33001)]) and show some applications to problems with strong nonclassical weights on \((0,+\infty)\), including a conjecture for an oscillatory weight on \([-1,1]\). Finally, we give some new results on orthogonal polynomials on radial rays in the complex plane.

MSC:

33C47 Other special orthogonal polynomials and functions
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
41A55 Approximate quadratures
65D30 Numerical integration
65D32 Numerical quadrature and cubature formulas

Citations:

Zbl 1081.33001
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