On a result of Fel'dman on linear forms in the values of some \(E\)-functions (Q1733359)

Let $m$ be a positive integer and $\mathbb K$ be $\mathbb Q$ or an imaginary quadratic field. Assume that $\mathbb Z_{\mathbb K}$ is the ring of integers of $\mathbb K$ and $\alpha\in\mathbb K\setminus \{ 0\}$. Let $\lambda_1,\dots ,\lambda_m$ be rational numbers different from negative integers and such that $\lambda_i-\lambda_j$ is not integer for all $i,j\in \{ 1,\dots ,m\}$, $i\not= j$. Then there exist positive constants $H_0$, $d_0$, $d_1$ and $d_2$ depending only on $m, \lambda_1,\dots ,\lambda_m$ and $\alpha$ such that for every $(\beta_0,\dots .\beta_m)\in\mathbb Z_{\mathbb K}^{m+1}$ with $H=\prod_{i=1}^m \max (1,\mid \beta_i\mid)\geq H_0$ we have \[ \left| \beta_0+\sum_{j=1}^m\beta_j\sum_{k=0}^\infty \frac{\alpha^k}{\prod_{i=1}^k(i+\lambda_j)}\right| >\frac 1{H^{1+\frac {6(d_0+d_1m+d_2m^2)}{\log\log H}}} . \] The proof is based on Baker type bounds and uses Padé approximations.