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Source. Peter B. Borwein, On the irrationality of certain series, Mathematical Proceedings of the Cambridge Philosophical Society 112(1) (1992), 141--146, doi:10.1017/s030500410007081x. Theorem 1 is stated on p. 142; its proof runs through Lemmas 1--5 and the proof of the theorem, pp. 142--144. Bibliographic details are on the source card.

Statement

Let $q$ be an integer with $|q|>1$ and let $c$ be a nonzero rational number with $c\ne-q^n$ for every $n$ (the paper states this exclusion as a standing assumption in parentheses after the theorem). Then

$$\sum_{n=1}^{\infty}\frac{1}{q^n+c}$$

is irrational.

The abstract (p. 141) adds that the series is not a Liouville number; that stronger claim is argued only in the unnumbered closing paragraph on p. 146, as a sketch, and is not part of Theorem 1.

The introduction (p. 141) says Theorem 1 extends the main theorem of the author's 1991 paper (J. Number Theory 37 (1991), 253--259, its reference [2]), which handles only $q>0$.

Proof sketch (pp. 142--144)

The proof works with the auxiliary series $S(c,q)=\sum_{h\ge1}1/(1-cq^h)$, where this $c$ is the proof's own parameter; replacing the theorem's $c$ by $-1/c$ turns the theorem's series into a rational multiple of $S$ (a step the paper leaves implicit). Shifting the proof's $c$ to $cq^m$ changes $S$ by a finite rational sum (the paper's equation (2), p. 142), so the proof may assume $|c|>2$ (p. 143).

• A contour integral $F_n(q)$ over $|t|=1$ (equation (1), p. 142) is evaluated by residues as a polynomial multiple of $S(c,q)$ plus a term from the pole at $t=0$ (Lemma 1, p. 142).
• The coefficient polynomial $p_n(c,q)$ has integer coefficients and degree $n-1$ in $c$, by a $q$-binomial identity derived from the Cauchy binomial theorem (Lemma 2, p. 143). After multiplying by $(n-2)!\prod_{k=1}^{n}(1-cq^k)\prod_{k=[n/2]}^{n}(1-q^k)$ the remaining term is a polynomial $s_n(c,q)$ with integer coefficients and degree at most $2n$ in $c$ (Lemma 3, p. 143).
• For $|q|\ge2$ and $|c|\ge2$, $|F_n(q)|\le 2^{n+1}/q^{3n^2/2}$ as printed, by moving the contour outward through the poles $t=cq^m$ (Lemma 4, statement p. 143, proof p. 144).
• For $|q|\ge2$ and $|c|\ge1$, $F_n(q)\ne0$ for all $n\ge n_0$: the residue terms are of one sign or alternate and decrease in modulus (Lemma 5, p. 144).
• Writing the proof's $c=\alpha/\beta$ and multiplying by $\beta^{2n}$ gives nonzero integer linear forms in $S$ that tend to zero, so $S$ is irrational (p. 144).

This sketch is written from a reading of the proof's structure; the estimates were not re-derived here.

Read depth. Claims checked: the statement was read clause by clause on p. 142 of the printed article; the proof was read for structure only.

Dependencies

Lemmas 1--5 of the paper (pp. 142--144); the Cauchy binomial theorem, which the paper cites from J. M. Borwein and P. B. Borwein, Pi and the AGM (Wiley, 1987), p. 76.

Bears on

• Problem 1050: the case $q=2$, $c=-3$ is the problem's series, so the theorem gives a second proof of its irrationality, after the 1991 paper.
• Problem 257: the case $q=2$, $c=-1$ gives the irrationality of $\sum_{n\ge1}1/(2^n-1)$, the set of all positive integers. With $q=2^d$ and $c=-2^{-a}$ the theorem gives every set $\{a+dk:k\ge0\}$ with $a,d\ge1$, and finitely modifying a set changes the sum by a rational number; the source card works this specialization out. The paper does not state these cases, and it says nothing about a general infinite set.
• Problem 264: context only. The theorem treats a constant shift of $q^n$; it says nothing about factorials, and it does not give the problem's predicate for $2^n$, which quantifies over every bounded nonzero integer sequence of shifts.