Inequalities and monotonicity properties for zeros of Hermite functions

Abstract
We study the variation of the zeros of the Hermite function H‚(t) with respect to the positive real variable ‚. We show that, for each nonnegative integer n, H‚(t) has exactly n + 1 real zeros when n < ‚ • n + 1 and that each zero increases from ¡1 to 1 as ‚ increases. We establish a formula for the derivative of a zero with respect to the parameter ‚; this derivative is a completely monotonic function of ‚. By-products include some results on the regular sign behaviour of difierences 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 H‚(t). Similar results on zeros of certain con∞uent hypergeometric functions are given too. These specialize to results on the flrst, second, etc., positive zeros of Hermite polynomials.

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