engleski

Zeros of Bessel function derivatives

We prove that for \nu>n-1 all zeros of the nth derivative of the Bessel function of the first kind J_\nu are real. Moreover, we show that the positive zeros of the nth and (n + 1)th derivative of the Bessel function of the first kind J_\nu are interlacing when \nu \ge n and n is a natural number or zero. Our methods include the Weierstrassian representation of the nth derivative, properties of the Laguerre-P\'olya 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 H_\nu are also proved. The main results obtained in this paper generalize and complement some classical results on the zeros of Bessel and Struve functions of the first kind. Some open problems related to Hurwitz' theorem on the zeros of Bessel functions are also proposed.

Zeros of Bessel and Struve functions ; Laguerre-Polya class of entire functions ; interlacing of positive zeros ; reality of the zeros ; Laguerre inequality ; Jensen polynomials ; Laguerre polynomials ; Rayleigh sums.

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