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Zbl 0843.33001
Lorch, Lee
The zeros of the third derivative of Bessel functions of order less than one.
(English)
[J] Methods Appl. Anal. 2, No.2, 147-159 (1995). ISSN 1073-2772

Main results: If $\lambda_1$, $\lambda_2$ are zeros of $J'''_\nu(x)$, $0\leq \nu< 1$, $\lambda_1< \lambda_2< j_{\nu 1}$, then $\lambda_1$ is a steadily decreasing function of $\nu$ as $\nu$ increases to 1 and $\lambda_2$ is a steadily increasing function of $\nu$ as $\nu$ increases to 1. Further, there exists a (unique) value of $\nu= \nu_0$ such that $J'''_\nu(x)$ has two zeros when $\nu_0< \nu< 1$ and none when $0< \nu< \nu_0$. When $\nu= \nu_0$, $J'''_\nu(x)$ has a douoble zero in $0< x< j_{\nu 1}$;'' Let $\lambda_1 (\nu)< \lambda_2 (\nu)$ denote the zeros of $J'''_\nu(x)$ in $0< x< j_{\nu 1}$, $\nu_0< \nu< 1$. Then $\lambda_1 (\nu) \downarrow 0$, $\lambda_2 (\nu) \uparrow \sqrt {3}= j'''_{11}$ as $\nu \uparrow 1$'' (Theorems 4.1 and 4.2 in the paper, respectively). Some inequalities for the zeros studied are also established. Here $\nu_0= 0.755578\dots$; $j_{\nu 1}$ is, as usual, the first positive zero of the Bessel function $J_\nu (x)$ and $j'''_{\nu k}$ $(k= 1, 2, \dots)$ stands for the positive zeros of the third derivative $J'''_\nu(x)$ and $J_\nu(x)$. The above results complete other ones found in another paper by the author and {\it P. SzegĂ¶} [Methods Appl. Anal. 2, 103-111 (1995; Zbl 0833.33003)].
[N.Hayek (La Laguna)]
MSC 2000:
*33C10 Cylinder functions, etc.

Keywords: monotonicity; inequalities; Bessel function; zeros

Citations: Zbl 0833.33003

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