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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 νk '' , k=1,2,··. of Bessel function J ν (x) of the first kind. They proved that j ν1 '' increases with ν>0 and that j νk '' increases on 0<ν<3838, for k=2,3,···. In the present paper the authors prove that j νk '' increases with ν10 for k=2,3,···. Their results together with Lorch and Szego results show that j νk '' increases with ν0, 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.


MSC:
33C10Bessel and Airy functions, cylinder functions, 0 F 1
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)