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Source. Theorem 3.1 and its proof, preprint p. 8; the quantities $Q_0$, $Q_1$ from Lemma 3.1 (pp. 6--7). Read on the rendered pages.

Statement

Fix integers $a>0$ and $b$ with $an+b\ne0$ for all $n\in\mathbb{N}$, and a polynomial $P(x)=\sum_{i=0}^Ta_ix^i\in\mathbb{Z}[x]$. The sum

$$R^*:=\sum_{N=1}^{\infty}\frac{P(N)}{\prod_{n=1}^{N}(an+b)}$$

is rational exactly when

$$Q_1=\sum_{i=0}^{T}a_i\sum_{k=0}^{i}\frac{1}{k!\,a^k} \sum_{j=0}^{k}(-1)^j\binom kj\Bigl(-\frac ba+k-j\Bigr)^i=0 .\qquad(17)$$

Proof pointer

p. 8: Lemma 3.1 (pp. 6--8) writes $R^*=Q_0+Q_1\sum_{N\ge1}1/\prod_{n\le N}(an+b)$ with rational $Q_0$, $Q_1$, and Oppenheim's criterion (Lemma 2.2) makes the last sum irrational, so $R^*$ is rational exactly when $Q_1=0$.

Consequences on the same page

Corollary 3.1 is the case $a=1$, $b=0$. Corollary 3.2: if $a\nmid a_T(b-1)^T$ then $R^*$ is irrational. Theorem 3.2: if $f:\mathbb{N}\to\mathbb{Z}$ satisfies $f(N)=P(N)+o(N)$ with $P\in\mathbb{Q}[x]$ and $\sum f(N)/\prod_{n\le N}(an+b)\in\mathbb{Q}$, then $f(N)=P(N)-Q_1$, printed "for all $N$" (see that page for the reading).

Bears on. No catalog problem directly; context for #252 through Corollary 3.1.