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Monotonicity properties of the zeros of Freud and sub-range Freud polynomials: Analytic and empirical results

Author: Walter Gautschi
Journal: Math. Comp. 86 (2017), 855-864
MSC (2010): Primary 33C47, 33F05
Published electronically: June 29, 2016
MathSciNet review: 3584551
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Abstract: Freud and sub-range Freud polynomials are orthogonal with respect to the weight function $ w(t)=\vert t\vert^\mu \exp (-\vert t\vert^\nu )$, $ \mu >-1$, $ \nu >0$, supported on the whole real line $ \mathbb{R}$, resp. on strict subintervals thereof. The zeros of these polynomials are studied here as functions of $ \nu $ and shown, analytically and empirically by computation, to collectively increase or decrease on appropriate intervals of the variable $ \nu $.

References [Enhancements On Off] (What's this?)

  • [1] Walter Gautschi, Orthogonal Polynomials: Computation and Approximation, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2004. MR 2061539
  • [2] W. Gautschi, Orthogonal polynomials in Matlab: Exercises and solutions, Software, Environments, Tools, SIAM, Philadelphia, PA, 2016.
  • [3] Gábor Szegő, Orthogonal Polynomials, 4th ed., Colloquium Publications, vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517

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Additional Information

Walter Gautschi
Affiliation: Department of Computer Science, Purdue University, West Lafayette, Indiana 47907-2066

Keywords: Freud polynomials, sub-range Freud polynomials, zeros of orthogonal polynomials, Matlab software
Received by editor(s): September 8, 2015
Published electronically: June 29, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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