Two theorems on meromorphic functions used as a principle for proofs on irrationality (Q1826300)

Building on \textit{E. M. Nikishin's} earlier work [Math. Notes 29, 270-271 (1981); translation from Mat. Zametki 29, 525-527 (1981; Zbl 0481.10035)], the authors prove the following result. Theorem. Let \(f\) be a transcendental entire function of strict order \(\leq \sigma<3/2.\) All derivatives of \(f\) at the points 0 and \(\lambda\neq 0\), as well as \(\lambda\) shall be numbers of \(\mathbb{Q}(i).\) Let \(D_0(n)\) and \(D_{\lambda}(n)\) be the least common multiple of the denominators of the numbers \(f^{(k)}(0)\) and \(f^{(k)}(\lambda),\) \(k=0, \ldots, n.\) Then for infinitely many \(n\in\mathbb{N},\) \[ D_0(2n)D_{\lambda}(2n)\geq C^n n^{\delta(\sigma)n}, \] where \(\delta(\sigma)=(2/\sigma)-1\) if \(0<\sigma\leq 1\), \(\delta(\sigma)=(3/\sigma)-2\) if \(1\leq \sigma <3/2\), and \(C\) does not depend on \(n\). The proof uses the diagonal Padé approximants of the exponential in an unusual and interesting way. The authors then apply their theorem to give simple proofs of irrationality results for various \(E\)-functions.