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Source. Jaroslav Hančl and Robert Tijdeman, On the irrationality of factorial series, Acta Arith. 118 (2005), 383--401; Theorem 3.3, preprint p. 9, proof pp. 9--10; Corollary 3.4 and the Remark after it, p. 10. Page numbers are those of the preprint named on the source card.

Statement

Theorem 3.3 (p. 9): "Let $a>0$ and $b$ be integers such that $an+b\ne0$ for every $n\in\mathbb{N}$. Let $P(x)=\sum_{i=0}^Ta_ix^i\in\mathbb{R}[x]$. Let $f:\mathbb{N}\to\mathbb{Z}$ be a sequence such that $f(N)=(aN+b)P(N)+O(1)$ as $N\to\infty$. Suppose $\sum_{N=1}^{\infty}\frac{f(N)}{\prod_{n=1}^N(an+b)}\in\mathbb{Q}$. Then $a_T,a_{T-1},\ldots,a_1,a_0\in\mathbb{Q}$."

Read depth. Claims checked: the statement was read clause by clause on the rendered page; the proof was read for structure only. Nothing here is independently reviewed.

Proof pointer

pp. 9--10: assuming the sum is $p/q$, take the largest index $U$ with $a_U\notin\mathbb{Q}$, remove the rational top part of $P$ with Lemma 3.1, and apply the $U$-th difference of the integers $R^*_N$ of Lemma 2.1 with Lemma 2.5; the result is an integer equal to a fixed nonzero multiple of $a_U$ (irrational) plus $O(1/N)$, which is impossible for large $N$.

Consequence on p. 10

Corollary 3.4: $\sum[P(N)]/N!$ is irrational for a real polynomial with nonnegative coefficients and positive leading coefficient.

Bears on. No catalog problem directly.