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Source. Jaroslav Hančl and Robert Tijdeman, On the irrationality of factorial series, Acta Arith. 118 (2005), 383--401; Theorem 3.2, preprint p. 8, proof p. 9; Corollary 3.3, p. 9. Page numbers are those of the preprint named on the source card.

Statement

Let $a>0$ and $b$ be integers with $an+b\ne0$ for every $n\in\mathbb{N}$, and let $P(x)=\sum_{i=0}^Ta_ix^i\in\mathbb{Q}[x]$. Let $f:\mathbb{N}\to\mathbb{Z}$ satisfy $f(N)=P(N)+o(N)$ as $N\to\infty$, and suppose

$$\sum_{N=1}^{\infty}\frac{f(N)}{\prod_{n=1}^N(an+b)}\in\mathbb{Q}.$$

Then, as printed, "$f(N)=P(N)-Q_1$ for all $N$", where $Q_1$ is the rational number given by (13) of Lemma 3.1 (p. 7): with $d$ the least common denominator of $a_0,\ldots,a_T$,

$$a^TdQ_1=\sum_{i=0}^{T}a_i\sum_{k=0}^{i}\sum_{h=k}^{i}\binom ih a^{T-i+h-k}(-b)^{i-h}S(h,k),$$

an integer, with $S(h,k)$ the Stirling numbers of the second kind of Lemma 2.3.

Corollary 3.3 (p. 9): under the conditions of Theorem 3.2, $P(N)\equiv Q_1\bmod 1$ for all $N$, and therefore $dQ_1\in\mathbb{Z}$.

Filing observation. The proof (p. 9) ends with $Q_1+f(N)-P(N)=0$ for $N\ge N_0$, and the conclusion cannot hold for all $N$ in general: changing $f(1)$ by any integer changes the sum by a rational number and leaves $f(N)=P(N)+o(N)$. The statement is read here as $f(N)=P(N)-Q_1$ for all sufficiently large $N$. This is a reading of the print, not a review verdict.

Read depth. Claims checked: the statement, Lemma 3.1 and Corollary 3.3 were read clause by clause on the rendered pages; the proof was read for structure only. Nothing here is independently reviewed.

Proof pointer

p. 9: Lemma 3.1 splits off the polynomial part, and Oppenheim's criterion (Lemma 2.2) applied to the remaining numerators $Q_1+f(N)-P(N)$, which are $o(N)$ with a fixed denominator, forces them to vanish from some point on.

Relation to Erdős problems

Corollary 3.4 uses this theorem with Theorem 3.3. The numerators $\sigma_k(n)$ of Problem 252 are not of the form $P(N)+o(N)$ for $k\ge1$, so the theorem does not apply to that series.

Bears on. No catalog problem directly.