Inequalities and monotonicity properties for zeros of Hermite functions

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 nonnegative 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. AMS Subject Classifications: Primary 33C15, 33C45; Secondary 34C10