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Interlacing properties of the zeros of Bessel functions. (English) Zbl 0811.33003

For k=1,2, denote, as usual, by j ν,k , y ν,k , j ν,k ' , y ν,k ' , the k-th positive zeros of the Bessel functions J ν (x), Y ν (x) of the first and second kind and their derivatives J ν ' (x), Y ν ' (x), respectively. Let c ν,k be the k-th positive zero of the cylinder function C ν (x)=cosαJ ν (x)-sinαY ν (x), ν>0, 0α<π. By using the notation introduced in two earlier papers [SIAM, J. Math. Anal. 15, 206-212 (1984; Zbl 0541.33001); Stud. Sci. Math. Hung. 25, 377-385 (1990; Zbl 0748.33002)], the authors define j ν,κ by j ν,κ =c ν,k , with κ=k-α/π, and similarly j ν,κ ' to denote the zeros of C ν ' (x) defined in such a way that (*) j ν,κ-1 <j ν,κ ' <j ν,κ . In the present paper they prove the following results:

Theorem 1. For the zeros j ν,κ of the cylinder function C ν (x) (ν>-κ) the inequalities j ν+1,κ 1 2(j ν,κ +j ν,κ+1 ), ν-1 2, κ1 2 hold.

Theorem 2. For the zeros j ν,κ of the cylinder function C ν (x) and for the zeros j ν,κ ' of C ν ' (x) satisfying the relation (*), the inequality j ν,κ-1/2 >j ν,κ ' for ν0, κ1 holds.

MSC:
33C10Bessel and Airy functions, cylinder functions, 0 F 1