## 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

### Keywords:

cylinder functions

Zbl 0681.33008
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