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Ratios of Bessel functions and roots of $\alpha {J}_{v}\left(x\right)+x{J}_{v}^{\text{'}}\left(x\right)=0$. (English) Zbl 0938.33002

The author shows that $\zeta {J}_{\nu }\left(\nu \zeta \right)/{J}_{\nu +1}\left(\nu \zeta \right)$ is stricty decreasing in $\nu$ on any interval not containing a singularity and obtaining an upper bound on its derivative:

$\frac{\partial }{\partial \nu }\left[\frac{{J}_{\nu }\left(\nu \zeta \right)}{{J}_{\nu +1}\left(\nu \zeta \right)}\right]\le -\frac{2}{{\nu }^{2}}·$

He also shows that $x\frac{\partial }{\partial v}\left[\frac{J\nu \left(x\right)}{{J}_{\nu +1}\left(x\right)}\right]\ge 2$ where $x{J}_{\nu }\left(x\right)/J\nu +1\left(x\right)$ is strictly increasing in $\nu$. The graph of ${J}_{\nu }\left(x\right)/{J}_{\nu +1}\left(x\right)$ as a function of $x$ is illustrated in figures for various values of $\nu$. These results generalize and sharpen previously known results, and allow the deduction of a more complete description, including monotonicity and multiplicity, of the positive roots of the equation $\alpha J\nu \left(x\right)+x{J}_{\nu }^{\text{'}}\left(x\right)=0$ for all real $\nu$ and $\alpha$.

##### MSC:
 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$
Bessel functions