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On the points of inflection of Bessel functions of positive order. I. (English) Zbl 0716.33003
The authors prove the following integral formula $$ \frac{dj''}{d\nu}=\frac{2\nu}{(j'')\sp 2J\sb{\nu}(j'')J'''\sb{\nu}(j'')}\{\int\sp{j''}\sb{0}\frac{J\sp 2\sb{\nu}(t)}{t}dt-J\sp 2\sb{\nu}(j'')\}, $$ where $j''=j''\sb{\nu k}$ is the kth zero of the second derivative of the Bessel function $J\sb{\nu}(x)$ of the first kind. They use this representation to investigate the variation of $j''\sb{\nu k}$ with respect to the order $\nu$, showing that $j''\sb{\nu 1}$ increases for $0<\nu <\infty$ and that $j''\sb{\nu k}$ increases in $0<\nu \le 3838$ for any fixed $k=2,3,... $. A monotonic result for $J\sb{\nu}(j''\sb{\nu k})$ is also established. Finally a conjecture on the completely monotonicity of some sequences involving the inflection points of the general Bessel function is formulated.
Reviewer: A.Laforgia

33C10Bessel and Airy functions, cylinder functions, ${}_0F_1$
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