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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

dj '' dν=2ν (j '' ) 2 J ν (j '' )J ν ''' (j '' ){ 0 j '' J ν 2 (t) tdt-J ν 2 (j '' )},

where j '' =j νk '' is the kth zero of the second derivative of the Bessel function J ν (x) of the first kind. They use this representation to investigate the variation of j νk '' with respect to the order ν, showing that j ν1 '' increases for 0<ν< and that j νk '' increases in 0<ν3838 for any fixed k=2,3,···. A monotonic result for J ν (j νk '' ) 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

MSC:
33C10Bessel and Airy functions, cylinder functions, 0 F 1