Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-22T18:36:46.033Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

1. Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions, National Bureau of Standards, 1964.Google Scholar
2. Borwein, P.B. and Erd, T.élyi, Polynomials and Polynomial Inequalities, Springer-Verlag, to appear.Google Scholar
3. Craven, T. and Csordas, G., The Gauss-Lucas Theorem and Jensen Polynomials, Trans. Amer. Math. Soc. 278(1983), 415–429.Google Scholar
4. Dieudonn, J.é, Sur les zéros des polynômes-sections de ex, Bull. Sci. Math. 70(1935), 333–351.Google Scholar
5. Dilcher, K. and Stolarsky, K.B., Sequences of polynomials whose zeros lie on fixed lemniscates, Period. Math. Hungar. 25(1992), 179–190.Google Scholar
6. Fomenko, S.V., On the zeros of partial sums of series of functions, Siberian Math. J. 10(1969), 296–306.Google Scholar
7. Grosswald, E., Bessel Polynomials, Lecture Notes in Math. 698, Springer-Verlag, Berlin, Heidelberg, New York, 1978.Google Scholar
8. Marden, M., Geometry of Polynomials, Amer. Math. Soc., Providence, Rhode Island, 1966.Google Scholar
9. P, G.ólya and Szeg, G.ö, Problems and Theorems in Analysis I, II, Springer-Verlag, Berlin, Heidelberg, New York, 1978.Google Scholar
10. Prather, C.L., Zeros of operators on functions and their analytic character, Rocky Mountain J. Math. 14(1984), 679–697.Google Scholar
11. Prather, C.L. and Shaw, J.K., Zeros of successive iterates of multiplier-sequence operators, Pacific J. Math. (1)104(1983), 205–218.Google Scholar
12. Rosenbloom, P.C., Distribution of zeros of polynomials, In: Lectures on Functions of a Complex Variable, (ed. Kaplan, W.), Univ. of Michigan Press, Ann Arbor, 1955. 265–285.Google Scholar
13. Saff, E.B. and Varga, R.S., On the zeros and poles ofPadé approximants to ez, Numer. Math. 25(1975), 1–14.Google Scholar
14. Szeg, G.ö, Über die Nullstellen von Polynomen, die in einem Kreise gleichmafiig konvergieren, Sitzungsber. Berlin Math. Ges. 21(1922), 59–64. Also in: Collected Papers I, 537–543.Google Scholar
15. Szeg, G.ö, Über eine Eigenschaft der Exponentialreihe, Sitzungsber. Berlin Math. Ges. 23(1924), 50–64. Also in: Collected Papers I, 646–660.Google Scholar
16. Varga, R. S., Scientific Computation on Mathematical Problems and Conjectures, Society for Industrial and Applied Mathematics, Philadelphia, 1990.Google Scholar
17. Walsh, J. L., The Location of Critical Points of Analytic and Harmonic Functions, Amer. Math. Soc, Providence, Rhode Island, 1950.Google Scholar