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Some monotonicity properties associated with the zeros of Bessel functions. (English) Zbl 0731.33002

Supplementing and extending results of A. Laforgia and the reviewer [Z. Angew. Math. Phys. 39, No.2, 267-271 (1988; Zbl 0681.33008)], the author proves some monotonicity results for \(a_{\alpha,k}\) and \((\alpha +2)a_{\alpha,k}\) where \(a_{\alpha,k}\) is the kth positive zero of a solution of the generalized Airy equation \(y''-x^{\alpha}y=0\). The method is to use \(a_{\alpha,k}=[c_{\nu k}/(2\nu)]^{2\nu}\) where \(\nu =1/(\alpha +2)\) and \(c_{\nu k}\) are the positive zero of cylinder functions of order \(\nu\). The results are expressed in terms of the latter zeros. They depend heavily on known results for \(c_{\nu k}\) including a formula due to G. N. Watson for \(dc_{\nu k}/d\nu\).

MSC:

33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

Citations:

Zbl 0681.33008
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