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Zeros of Iterated Integrals of Polynomials

Published online by Cambridge University Press:  20 November 2018

Peter B. Borwein ,
Weiyu Chen  and
Karl Dilcher
Peter B. Borwein
Affiliation:
Department of Mathematics Simon Fraser University Burnaby, British Columbia V5A 1S6
Weiyu Chen
Affiliation:
Department of Mathematics University of Alberta Edmonton, Alberta T6G 2G1
Karl Dilcher
Affiliation:
Department of Mathematics, Statistics and Computing Science Dalhousie University Halifax, Nova Scotia B3H3J5
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Abstract

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The operator Im is defined as m-fold indefinite integration with zero constants of integration. The zero distribution of Im(p) for polynomials p is studied in general, and for two special classes of polynomials in detail. The main results are: (i) The zeros of In(Pn), where Pn(𝑧) is the n-th Legendre polynomial, converge to a certain algebraic curve; (ii) the zeros of an integer) converge to pieces of a circle and of two "Szegö curves".

Keywords

30C1530E1533C45Zeros of polynomialsiterated integralsGauss-Lucas theoremSzegö curveLegendre polynomials
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 1 , 01 February 1995 , pp. 65 - 87
Copyright
Copyright © Canadian Mathematical Society 1995

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