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On the points of inflection of Bessel functions of positive order. II. (English) Zbl 0731.33001
[For part I see the authors in Proc. R. Soc. Lond., Ser. A 431, No.1883, 509-518 (1990; Zbl 0719.33001).] L. Lorch and P. Szego have established some monotonicity results on the inflection points $j''\sb{\nu k}$, $k=1,2,..$. of Bessel function $J\sb{\nu}(x)$ of the first kind. They proved that $j''\sb{\nu 1}$ increases with $\nu >0$ and that $j''\sb{\nu k}$ increases on $0<\nu <3838$, for $k=2,3,... $. In the present paper the authors prove that $j''\sb{\nu k}$ increases with $\nu\ge 10$ for $k=2,3,... $. Their results together with Lorch and Szego results show that $j''\sb{\nu k}$ increases with $\nu\ge 0$, for any $k=1,2,... $. The authors obtain the results with a sophisticated use, with error estimates, of asymptotic approximations. Some of these approximations are due to Olver.

33C10Bessel and Airy functions, cylinder functions, ${}_0F_1$
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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