Zeros of Bessel function derivatives
Copyright policy )Zeros of Bessel function derivatives
We prove that for $ν>n-1$ all zeros of the $n$th derivative of Bessel function of the first kind $J_ν$ are real and simple. Moreover, we show that the positive zeros of the $n$th and $(n+1)$th derivative of Bessel function of the first kind $J_ν$ are interlacing when $ν\geq n,$ and $n$ is a natural number or zero. Our methods include the Weierstrassian representation of the $n$th derivative, properties of the Laguerre-Pólya class of entire functions, and the Laguerre inequalities. Some similar results for the zeros of the first and second derivative of the Struve function of the first kind $\mathbf{H}_ν$ are also proved. The main results obtained in this paper generalize and complement some classical results on the zeros of Bessel functions of the first kind. Some open problems related to Hurwitz theorem on the zeros of Bessel functions are also proposed, which may be of interest for further research.
10 pages
- Óbuda University Hungary
- Far East Branch of the Russian Academy of Sciences Russian Federation
- SVEUCILISTE U ZAGREBU FAKULTET ELEKTROTEHNIKE I RACUNARSTVA Croatia
- Institute of Applied Mathematics Russian Federation
- University of Patras Greece
Representations of entire functions of one complex variable by series and integrals, interlacing of positive zeros, zeros of Bessel and Struve functions, Rayleigh sums., Rayleigh sum, Laguerre inequality, Mathematics - Classical Analysis and ODEs, Hurwitz's theorem, Zeros of Bessel and Struve functions ; Laguerre-P olya class of entire functions ; interlacing of positive zeros ; reality of the zeros ; Laguerre inequality ; Jensen polynomials ; Laguerre polynomials ; Rayleigh sums., Laguerre-Polya class of entire functions, Jensen polynomials, derivatives, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Laguerre polynomials, Bessel and Airy functions, cylinder functions, \({}_0F_1\), Laguerre-Pólya class, 33C10, 30D15, Zeros of Bessel and Struve functions, reality of the zeros
Representations of entire functions of one complex variable by series and integrals, interlacing of positive zeros, zeros of Bessel and Struve functions, Rayleigh sums., Rayleigh sum, Laguerre inequality, Mathematics - Classical Analysis and ODEs, Hurwitz's theorem, Zeros of Bessel and Struve functions ; Laguerre-P olya class of entire functions ; interlacing of positive zeros ; reality of the zeros ; Laguerre inequality ; Jensen polynomials ; Laguerre polynomials ; Rayleigh sums., Laguerre-Polya class of entire functions, Jensen polynomials, derivatives, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Laguerre polynomials, Bessel and Airy functions, cylinder functions, \({}_0F_1\), Laguerre-Pólya class, 33C10, 30D15, Zeros of Bessel and Struve functions, reality of the zeros
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