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Source. Jaroslav Hančl and Robert Tijdeman, On the irrationality of factorial series, Acta Arith. 118 (2005), 383--401; Theorem 3.4, preprint p. 10, proof p. 11; Corollaries 3.5 and 3.6, p. 11. Page numbers are those of the preprint named on the source card.

Statement

Let $K\ge0$, $a>0$ 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, as $N\to\infty$,

$$F(N+j)=\sum_{r=0}^{K}\frac{F^{(r)}(N)}{r!}j^r+O\Bigl(\frac{F(N)}{N^{K+1}}\Bigr) \quad\text{for }j=0,1,\ldots,K,\qquad(18)$$ $$F^{(r)}(N)=O\Bigl(\frac{F(N)}{N^r}\Bigr)\quad\text{for }r=0,1,\ldots,K,\qquad(19)$$ $$\lim_{N\to\infty}F^{(K)}(N)=0,\qquad \lim_{N\to\infty}\frac{N^{K+1}|F^{(K)}(N)|}{F(N)}=\infty,\qquad(20)$$

and

$$\limsup_{N\to\infty}N|F^{(K)}(N)|=\infty.\qquad(21)$$

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.

Read depth. Claims checked: the statement was read clause by clause on the rendered page; the proof was read for structure only. Lemma 2.5, on which it rests, was not read in full. Nothing here is independently reviewed.

Proof pointer

p. 11: assuming $R^*=p/q$, the $K$-th difference of the tails $R^*_N$ is $1/q$ times an integer by Lemma 2.1 and, by Lemma 2.5, equals $(-1)^KF^{(K)}(N)(1+o(1))+O(1/N)$; (20) makes it tend to $0$, so it vanishes for large $N$, and that contradicts (21).

Consequences on p. 11

• Corollary 3.5: $\sum[\gamma N^\alpha]/N!\notin\mathbb{Q}$ for $\alpha\ge0$, $\gamma>0$.
• Corollary 3.6: for $\alpha\in\mathbb{R}_{\ge0}\setminus\mathbb{Z}$ and $\gamma\in\mathbb{R}_+$, $\sum_{N\ge1}[\gamma N^\alpha\log N]/N!\notin\mathbb{Q}$ (Theorem 3.4 with $a=1$, $b=0$, $K=[\alpha]$, $F(N)=\gamma N^{\alpha-1}\log N$).

The paper notes (p. 11) that Theorem 3.4 does not reach $\sum[N\log N]/N!$, which is the reason for Theorem 3.5.

Relation to Erdős problems

The theorem needs a numerator that is, up to $O(1)$, $(aN+b)$ times a smooth function with the derivative conditions (18)--(21). The numerators $\sigma_k(n)$ of Problem 252 and $p_n^k$ of the factorial theorem discussed on Problem 251 are not given in that form, and the paper does not apply the theorem to them.

Bears on. No catalog problem directly.