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A note on higher monotonicity properties of generalized Airy functions. (English) Zbl 0585.33012

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

MSC:

33E99 Other special functions
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34A40 Differential inequalities involving functions of a single real variable
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References:

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