On the square of the zeros of Bessel functions. (English) Zbl 0541.33001

For \(\nu \geq 0\), let \(c\equiv c(\nu,k,\alpha)\) be the kth positive x-zero of \[ J_{\nu}(x) \cos \alpha-Y_{\nu}(x) \sin \alpha,\quad 0\leq \alpha<\pi, \] and let \(c(\nu,k,\alpha)\) be continued analytically to the \(\nu\)-interval \((\alpha/\pi-k,0).\) The authors’ main result, in a different notation, is that there exists \(\kappa_ 0\), \(0<\kappa_ 0<1\), such that when \(k-\alpha/\pi >\kappa_ 0\) we have \(dc/d\nu>1\) and \(d^ 2(c^ 2)/d\nu^ 2>0\), \(0\leq \nu<\infty\). This implies that \(j^ 2_{\nu k}\) is convex on \(0\leq \nu<\infty\) where \(j_{\nu k}=c(\nu,k,0)\), \(k=1,2,..\).. The authors’ method is to consider \(c(\nu,k,\alpha)\) as the solution of \[ dc/d\nu =2c\int^{\infty}_{0}K_ 0(2c\quad \sinh \quad t)e^{-2\nu t}dt, \] which satisfies \(c\to 0\) as \(\nu \downarrow(\alpha /\pi-k)\). They show that \(j^ 2_{\nu k}\) fails to be convex on \((-k,\infty)\) for \(k=2,3,..\). but they conjecture that \(j^ 2_{\nu l}\) is convex on \((- 1,\infty)\).
Reviewer: M.Muldoon


33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
Full Text: DOI