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Interlacing of positive real zeros of Bessel functions. (English) Zbl 1208.33007
The positive zeros ${j}_{\nu ,s}$, ${y}_{\nu ,s}$, ${j}_{\nu ,s}^{\text{'}}$, ${y}_{\nu ,s}^{\text{'}}$, ${j}_{\nu +\epsilon ,s}$, ${y}_{\nu +\epsilon ,s}$, $s=1,2,\cdots$, of Bessel functions ${J}_{\nu }\left(x\right)$, ${Y}_{\nu }\left(x\right)$, ${J}_{\nu }^{\text{'}}\left(x\right)$, ${Y}_{\nu }^{\text{'}}\left(x\right)$, ${J}_{\nu +\epsilon }\left(x\right)$, ${Y}_{\nu +\epsilon }\left(x\right)$, respectively, interlace according the inequalities: ${j}_{\nu ,s}^{\text{'}}<{y}_{\nu ,s}<{y}_{\nu +\epsilon ,s}<{y}_{\nu ,s}^{\text{'}}<{j}_{\nu ,s}<{j}_{\nu +\epsilon ,s}<{j}_{\nu ,s+1}^{\text{'}}<\cdots$, $s=1,2,\cdots$, for $\nu \ge 0$ and $0<\epsilon \le 1$. For $\epsilon =1$, analogous results are widely known.
##### MSC:
 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$ 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
##### Keywords:
Bessel functions; zeros of Bessel functions
##### References:
 [1] Abramowitz, M.; Stegun, I. A.: Handbook of mathematical functions, (1972) · Zbl 0543.33001 · doi:http://www.cs.bham.ac.uk/~aps/research/projects/as/ [2] Pálmai, T.; Apagyi, B.: On nonsingular potentials of Cox-Thompson inversion scheme, J. math. Phys. 51, 022114 (2010) [3] Watson, G. N.: A treatise on the theory of Bessel functions, Cambridge math. Lib. (1995) [4] Liu, H. Y.; Zou, J.: Zeros of the Bessel and spherical Bessel functions and their applications for uniqueness in inverse acoustic obstacle scattering, IMA J. Appl. math. 72, 817-831 (2007) · Zbl 1138.35072 · doi:10.1093/imamat/hxm013 [5] H.Y. Liu, J. Zou, Zeros of the Bessel and spherical Bessel functions and their applications for uniqueness in inverse acoustic obstacle scattering problems, Technical Report CUHK-2007-02 (342), The Chinese University of Hong Kong, Hong Kong, 2007.