The zeros of the third derivative of Bessel functions of order less than one (Q1907523)

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 \textit{P. Szegö} [Methods Appl. Anal. 2, 103-111 (1995; Zbl 0833.33003)].