Transcendence measure for the values of the logarithmic derivative of the Bessel function $J_0$ at algebraic arguments

Let $P\in \mathbb{Z}[X]\setminus \{0\}$ be of degree $\delta \ge 1$ and usual height $H\ge 1$, and let $\alpha \in {\bar{\mathbb{Q}}}^{\ast }$ be of degree $d\ge 2$. As a consequence of general result due to Lang and Galochkin, we have the following transcendence measure: for any $\epsilon >0$, there exists $c>0$ such that $\mathrm{\mid }P(J_0^{\mathrm{\prime }}(\alpha )\mathrm{/}{J}_0(\alpha ))\mathrm{\mid }>c\mathrm{/}{H}^{4d^2\delta +\epsilon }$ where $J_0$ is the Bessel function. In this paper, we prove that the exponent $4d^2\delta$ can be replaced by a smaller (explicit) quantity $\mu (d,\delta )\le 4d^2\delta -2d\delta -1$. A similar improvement holds more generally for the logarithmic derivative of any $E$-function $f$ of differential order 2 and such that $f$ and $f^{\mathrm{\prime }}$ are homogeneously algebraically independent. Our method rests upon the optimization of the size of a determinant that appears naturally in the classical Siegel-Shidlovskii method, following the steps of our previous improvement of the transcendence measure of the value $e^{\alpha }$ for any $\alpha \in {\bar{\mathbb{Q}}}^{\ast }$.