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

Source abstract

Let $P\in \mathbb Z[X]\setminus\{0\}$ be of degree $δ\ge 1$ and usual height $H\ge 1$, and let $α\in \overline{\mathbb Q}^*$ 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 $\varepsilon>0$, there exists $c>0$ such that $\vert P(J_0'(α)/J_0(α))\vert>c/H^{4d^2δ+\varepsilon}$ where $J_0$ is the Bessel function. In this paper, we prove that the exponent $4d^2δ$ can be replaced by a smaller (explicit) quantity $μ(d,δ)\le 4d^2δ-2dδ-1$. A similar improvement holds more generally for the logarithmic derivative of any $E$-function $f$ of differential order 2 and such that $f$ and $f'$ 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 $e^α$ for any $α\in \overline{\mathbb Q}^*$.