Monotonicity properties of zeros of generalized Airy functions. (English) Zbl 0681.33008

Laforgia, Andrea; Muldoon, Martin E.

Z. Angew. Math. Phys. 39, No. 2, 267-271 (1988).

The generalized Airy function is a solution of the differential equation \[ (1)\quad y''+x^{\alpha}y=0,\quad x\in [0,\infty), \] where \(\alpha\) is a positive number. From the introduction: “M. S. P. Eastham conjectured that the first positive zero \(a_{\alpha 1}\) of a solution of (1) with \(y(0)=0\), decreases as \(\alpha\) increases. We show here that this decrease (to 1) occurs for all positive zeros of such a solution and indeed for all, except possibly the first of the zeros of any nontrivial solution of (1) even without the condition \(y(0)=0.\)”

Reviewer: L.Littlejohn

MSC:

33C10	Bessel and Airy functions, cylinder functions, \({}_0F_1\)
34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34B30	Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)

Keywords:

monotonicity of zeros; Airy function

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