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Polynomials orthogonal on the semicircle, II

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Abstract

Generalizing previous work [2], we study complex polynomials {π k },π k (z)=z k+⋯, orthogonal with respect to a complex-valued inner product (f,g)=∫ 0 π f(e iθ)g(e iθ)w(e iθ)dθ. Under suitable assumptions on the “weight function”w, we show that these polynomials exist whenever Re ∫ 0 π w(e iθ)dθ≠0, and we express them in terms of the real polynomials orthogonal with respect to the weight functionw(x). We also obtain the basic three-term recurrence relation. A detailed study is made of the polynomials {π k } in the case of the Jacobi weight functionw(z)=(1−z)α(1+z)β, α>−1, and its special case\(\alpha = \beta = \lambda - \tfrac{1}{2}\) (Gegenbauer weight). We show, in particular, that for Gegenbauer weights the zeros ofπ n are all simple and, ifn≥2, contained in the interior of the upper unit half disc. We strongly suspect that the same holds true for arbitrary Jacobi weights. Finally, for the Gegenbauer weight, we obtain a linear second-order differential equation forπ n (z). It has regular singular points atz=1, −1, ∞ (like Gegenbauer's equation) and an additional regular singular point on the negative imaginary axis, which depends onn.

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References

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    W. Gautschi (1959):Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. and Phys.,38:77–81.

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    W. Gautschi, G. V. Milovanović (1986):Polynomials orthogonal on the semicircle. J. Approx. Theory,46:230–250.

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    W. Gautschi, Jet Wimp (to appear):Computing the Hilbert transform of a Jacobi weight function. BIT.

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    B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. Ikebe, V. C. Klema, C. B. Moler (1976): Matrix Eigensystem Routines—EISPACK Guide, 2nd edn. Lecture Notes in Computer Science, vol. 6. New York: Springer-Verlag.

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Communicated by Paul Nevai.

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Gautschi, W., Landau, H.J. & Milovanović, G.V. Polynomials orthogonal on the semicircle, II. Constr. Approx 3, 389–404 (1987). https://doi.org/10.1007/BF01890577

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Key words and phrases

  • Complex orthogonal polynomials
  • Recurrence relations
  • Zeros
  • Differential equation

AMS classification

  • Primary 30C10
  • 30C15
  • 33A65
  • Secondary 34A30