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Zbl 0585.33012
Háčik, Miloš; Ivan, Peter
A note on higher monotonicity properties of generalized Airy functions. (English)
[J] Acta Univ. Palacki. Olomuc., Fac. Rerum Nat. 79, Math. 23, 93-101 (1984).

Let $\{x\sb k\}$ and $\{$ $x'\sb k\}$ be the sequence of positive zeros of y and $y'$ on (p,$\infty)$ where $$ (*)\quad y''+[\beta\sp 2\gamma\sp 2x\sp{2\beta -2}+(\beta\sp 2\nu\sp 2-1/4)x\sp{-2}]y=0 $$ $\gamma >0$, $1<\beta <3/2$, $\vert \nu \vert \ge 1/(2\beta)$ and $p\sp{2\beta}=(\beta\sp 2\nu\sp 2-1/4)/(\beta\sp 2\gamma\sp 2).$ The solutions may be expressed in terms of cylinder (Bessel) functions. The authors show the complete monotonicity of $\{x\sb k-x'\sb k\}$ (provided $x\sb 0>x'\sb 0)$, $\{y\sp 2(x'\sb k)\}$ and several other sequences involving zeros, zeros of derivative, etc. of a solution or a pair of solutions of (*).
[M.E.Muldoon]

MSC 2000:: *33E99 Special functions
34C10 Qualitative theory of oscillations of ODE: Zeros, etc.
34A40 Differential inequalities (ODE)

Keywords: generalized Airy functions; convexity; Bessel functions; zeros; monotonicity

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