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Asymptotic relation for zeros of cross-product of Bessel functions and applications. (English) Zbl 1416.33008
The cross-product of Bessel functions given by $f_{\nu,R}(x)=J_\nu(Rx) Y_\nu(x)-J_\nu(x) Y_\nu(Rx),$ where $$\nu, x\geq0$$ and $$R>1$$, has an infinite number of simple zeros. The $$k\/$$th positive zero of $$f_{\nu,R}(x)$$ is denoted by $$a_{\nu,k}$$, $$k=1, 2, \ldots\,$$. By using a known expression for $$da_{\nu,k}/d\nu$$, the author shows that $\lim_{k\to\infty}\frac{a_{kx,k}}{k}=\alpha(x)\qquad x\geq0,$ where $$\alpha(x)$$ is the unique solution of an initial value problem for a first-order differential equation. In addition, the upper bound $$a_{\nu,k}<\frac{\pi k}{R-1}+\frac{\pi\nu}{2R}$$ is established.
These results are applied to the determination of the Pleijel constant Pl for planar annular regions $$A_R:=\{x\in \mathbb{R}^2,1<|x|<R\}$$, $$R>1$$. This quantity describes the nodal domain statistics for the eigenfunctions of the Dirichlet problem $$\Delta u=-\lambda u$$ in $$\Omega$$, $$u=0$$ on $$\partial\Omega$$, where $$\Omega\in\mathbb{R}^2$$ is a bounded domain. It is shown that $\mathrm{Pl}(A_R)\geq\frac{8}{R^2-1} \sup_{x>0}\,\biggl\{\frac{x}{\alpha(x)^2}\biggr\}.$ Some numerical estimates of Pl($$A_R)$$ for different $$R>1$$ are given.

##### MSC:
 33C10 Bessel and Airy functions, cylinder functions, $${}_0F_1$$ 34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
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