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Asymptotics of zeros of polynomials arising from rational integrals

Authors
Journal
Journal of Mathematical Analysis and Applications
0022-247X
Publisher
Elsevier
Publication Date
Volume
299
Issue
1
Identifiers
DOI: 10.1016/j.jmaa.2004.06.016
Keywords
  • Rational Integral
  • Polynomial
  • Zeros
  • Asymptotic Behavior
  • Lemniscate

Abstract

Abstract We prove that the zeros of the polynomials P m ( a ) of degree m, defined by Boros and Moll via P m ( a ) = 2 m + 3 / 2 π ( a + 1 ) m + 1 / 2 ∫ 0 ∞ d x ( x 4 + 2 a x 2 + 1 ) m + 1 , approach the lemniscate { ζ ∈ C : | ζ 2 − 1 | = 1 , R ζ < 0 } , as m → ∞ .

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