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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 | References | Similar Articles | Additional Information

Abstract: Freud and sub-range Freud polynomials are orthogonal with respect to the weight function $w(t)=|t|^\mu \exp (-|t|^\nu )$, $\mu >-1$, $\nu >0$, 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?)

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

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Additional Information

Walter Gautschi
Affiliation: Department of Computer Science, Purdue University, West Lafayette, Indiana 47907-2066
MR Author ID: 71975

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