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Inequalities and monotonicity properties for zeros of Hermite functions

Published online by Cambridge University Press:  14 November 2011

Árpád Elbert  and
Martin E. Muldoon
Árpád Elbert
Affiliation:
Mathematical Institute, Hungarian Academy of Sciences, POB 127, H-1364 Budapest, Hungary
Martin E. Muldoon
Affiliation:
Department of Mathematics, York University, Toronto, Ontario, Canada, M3J 1P3
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Extract

We study the variation of the zeros of the Hermite function Hλ(t) with respect to the positive real variable λ. We show that, for each non-negative integer n, Hλ(t) has exactly n + 1 real zeros when n < λ ≤ n + 1, and that each zero increases from – ∞ to ∞ as λ increases. We establish a formula for the derivative of a zero with respect to the parameter λ; this derivative is a completely monotonic function of λ. By-products include some results on the regular sign behaviour of differences of zeros of Hermite polynomials as well as a proof of some inequalities, related to work of W. K. Hayman and E. L. Ortiz for the largest zero of Hλ(t). Similar results on zeros of certain confluent hypergeometric functions are given too. These specialize to results on the first, second, etc., positive zeros of Hermite polynomials.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 1 , 1999 , pp. 57 - 75
Copyright
Copyright © Royal Society of Edinburgh 1999

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