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Source. Jaroslav Hančl and Robert Tijdeman, On the irrationality of factorial series, Acta Arith. 118 (2005), 383--401; Theorem 3.5, preprint pp. 11--12, proof pp. 12--13; Corollaries 3.7 and 3.8, p. 13. Page numbers are those of the preprint named on the source card.

Statement

Let $K\ge1$, $a>0$ and $b$ be given integers with $an+b\ne0$ for every $n\in\mathbb{N}$. Let $F:\mathbb{R}_+\to\mathbb{R}_+$ be a function such that

$$F(N+x)=\sum_{r=0}^{\infty}\frac{F^{(r)}(N)}{r!}x^r\quad\text{for }x=o(N) \text{ as }N\to\infty,\qquad(22)$$ $$F^{(r)}(N)=O\Bigl(r!\,\frac{F(N)}{N^r}\Bigr)\quad\text{uniformly for } r=0,1,\ldots\text{ as }N\to\infty,\qquad(23)$$ $$\lim_{x\to\infty}F^{(K)}(x)=0,\qquad \lim_{x\to\infty}\frac{x^{K+1}|F^{(K)}(x)|}{F(x)}=\infty\qquad(24)$$

and

$$\lim_{x\to\infty}x^2|F^{(K)}(x)|=\infty.\qquad(25)$$

Let $f:\mathbb{N}\to\mathbb{Z}$ be a sequence such that $R^*:=\sum_{N=1}^{\infty}f(N)/\prod_{n=1}^N(an+b)$ is absolutely convergent and $f(N)=(aN+b)F(N)+O(1)$ as $N\to\infty$. Then $R^*$ is irrational.

Compared with Theorem 3.4, the paper says (p. 11), condition (21) becomes weaker while (18) and (19) become stronger; (22) and (23) imply (18) and (19) (p. 12).

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

Proof pointer

pp. 12--13: by Theorem 3.4 one may assume $NF^{(K)}(N)=O(1)$; the paper then compares the $(K-1)$-th differences at $N$ and at $N+t$ for a shift $t=o(N)$ chosen from (24) and (25), applies the mean value theorem, and finds an integer that tends to $0$ but whose vanishing contradicts (25).

Consequences on p. 13

• Corollary 3.7: let $\alpha\in\mathbb{R}_{\ge0}$, $\beta\in\mathbb{R}$, $\beta\ne0$, $\gamma\in\mathbb{Q}_+$, with $\beta>0$ whenever $\alpha=0$. Then $\sum_{N\ge1}[\gamma N^\alpha\log^\beta N]/N!\notin\mathbb{Q}$.
• Corollary 3.8: let $\alpha\in\mathbb{R}_{\ge0}$, $0<\beta<1$, $\gamma\in\mathbb{Q}_+$. Then $\sum_{N\ge1}[\gamma N^\alpha\exp(\log^\beta N)]/N!\notin\mathbb{Q}$.

In both corollaries $\gamma$ is restricted to positive rationals, unlike Corollaries 3.5 and 3.6, where $\gamma\in\mathbb{R}_+$.

Bears on. No catalog problem directly.