Bounds for the small real and purely imaginary zeros of Bessel and related functions
Denote by \(J_\nu (z)\) the Bessel function of order \(\nu\). Since \(z^{- \nu} J_\nu (z)\) is an even entire function its zeros are located symmetrically with respect to both the real and the imaginary axis. Denote by \(z_{\nu 1}\) the smallest positive zero of \(J_\nu (z)\) for \(\nu > - 1\). As \(\nu \to - 1 + \), \(z_{\nu 1}\) and \(- z_{\nu 1}\) collapse at the origin and for \(-1 > \nu < - 2\), they split again into two imaginary zeros \(z_{\nu 1}\) and \(- z_{\nu 1}\). The aim of this paper is to give bounds for \(z^2_{\nu 1} (< 0)\), \(- 2 < \nu < - 1\) and \(z^2_{\nu 1} (> 0)\), \(\nu > - 1\) in terms of \(\nu\). Two methods are used for this purpose. The first one is going back to Euler and Rayleigh and uses the power sums of the zeros and Rayleigh's inequalities, the second one is based on continued fractions of \({zJ_{\nu - 1} (z) \over J_\nu (z)}\) and Lommel polynomials \(R_{n, \nu}\). In order to give a flavour of the results proved, consider the following inequalities shown among others in the paper: \[ \begin{aligned} 2 (\nu + 1) (\nu + 3) & < z^2_{\nu 1} < 4 (\nu + 1) (\nu + 2) \text{ for } - 2 < \nu < - 1 \quad \text{and} \\ z^2_{\nu 1} & < x^2_{n, \nu + 1}, \text{ for } n \geq 2,\;\nu > - 1 \quad \text{and} \\ - y^2_{2n - 1, \nu + 1} & < z^2_{\nu 1} < - y^2_{2n, \nu + 1}, \text{ for } n \geq 2,\;- 2 < \nu < - 1, \end{aligned} \] where \(x_{n, \nu + 1}\), \(y_{n, \nu + 1}\) denote the smallest positive zero of \(R_{n, \nu + 1} (x)\) and \(R_{n, \nu + 1} (iy)\) respectively. Related functions like \(\alpha J_\nu (z) + zJ_\nu' (z)\), \(J_\nu' (z)\), \(J_{\nu}'' (z)\) are investigated using the same approaches.
scientific article; zbMATH DE number 798941