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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Bounds for zeros of some special functions

Author: Herbert W. Hethcote
Journal: Proc. Amer. Math. Soc. 25 (1970), 72-74
MSC: Primary 34.42; Secondary 33.00
MathSciNet review: 0255909
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Abstract | References | Similar Articles | Additional Information

Abstract: For $n \geqq 1$ let ${b_n}$ and ${c_n}$ be zeros (ordered by increasing values) of $u(x)$ and $v(x)$, respectively, which are non-trivial solutions of $u'' + p(x)u = 0$ and $v'' + q(x)v = 0$ with continuous $p(x)$ and $q(x)$. It is shown that if ${b_n} - {c_n} \to 0$ as $n \to \infty ,\;p(x) \geqq q(x)$, and either $p(x)$ or $q(x)$ is nonincreasing, then ${b_n} \geqq {c_n}$ for $n \geqq 1$. Inequalities related to asymptotic expansions are obtained for the negative zeros ${a_n}$ of the Airy function $Ai(z)$ and the zeros ${j_{v,n}}$ of the Bessel function ${J_v}(x)$.

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

  • E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944. MR 0010757
  • J. C. P. Miller, The Airy Integral, Giving Tables of Solutions of the Differential Equation $y”=xy$, Cambridge, at the University Press; New York, The Macmillan Company, 1946. MR 0018971
  • G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1939. MR 1, 14.
  • G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746

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Additional Information

Keywords: Zeros of special functions, asymptotic expansions, approximation of zeros, Sturm comparison theorem, Airy functions, Bessel functions
Article copyright: © Copyright 1970 American Mathematical Society