Borwein: On the irrationality of certain series

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theorem_1: For every integer q with absolute value above one and every nonzero rational c different from each -q^n, the sum over n of one over q^n plus c is irrational, extending the author's 1991 theorem from positive q to negative q.

theorem_2: For every integer q with absolute value above one and every nonzero rational c different from each -q^n, the alternating sum over n of (-1)^n over q^n plus c is irrational.

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The edition read for this card is the printed Math. Proc. Cambridge Philos. Soc. 112(1) article, 6 pages, pp. 141--146; page numbers below are the printed ones. The article prints only "Printed in Great Britain" on p. 141 and no copyright line; the journal's article page (https://www.cambridge.org/core/product/identifier/S030500410007081X/type/journal_article, read 2026-10-02) states "Copyright © Cambridge Philosophical Society 1992" and does not mark the article Open Access, every other right reserved.

Peter B. Borwein, "On the irrationality of certain series," Mathematical Proceedings of the Cambridge Philosophical Society, 112(1), 141-146, 1992. https://doi.org/10.1017/s030500410007081x

Overview

Borwein studies the two complete geometric-denominator series

$$U(q,r)=\sum_{n\ge 1}\frac1{q^n+r},\qquad V(q,r)=\sum_{n\ge 1}\frac{(-1)^n}{q^n+r},$$

where $q\in\mathbb Z$, $|q|>1$, and $r\in\mathbb Q\setminus\{0\}$, with the pole cases $r=-q^n$ excluded. The theorems name the shift $c$; this card writes $r$, the abstract's letter, and keeps $c$ for the proof's own parameter below. Theorem 1 (pp. 142–144) proves that $U(q,r)$ is irrational; it extends the author's earlier positive-$q$ result cited as [2]. Theorem 2 (pp. 145–146) proves the corresponding irrationality of $V(q,r)$. Thus the results concern complete series indexed by every positive integer, rather than arbitrary subseries. The scope is fixed rational shifts of geometric powers, with or without the prescribed alternating signs; arbitrary perturbations, arbitrary coefficient sequences, and nongeometric denominator sequences are not treated.

The proof of Theorem 1 is organized around the contour integral $F_n(q)$ in equation (1) (p. 142). For the auxiliary series

$$S(c,q)=\sum_{h\ge1}\frac1{1-cq^h},$$

Lemma 1 (p. 142) evaluates $F_n(q)$ by residues at $q^{-1},\ldots,q^{-n}$ and at zero, producing a linear expression in $S(c,q)$. Taking $c=-1/r$ gives $S(c,q)=rU(q,r)$, so $S$ is the normalized form of the target series. Equation (2) (p. 142),

$$S(cq^m,q)=S(c,q)-\sum_{h=1}^{m}\frac1{1-cq^h},$$

shows that multiplying $c$ by a power of $q$ changes the relevant number only by a rational finite sum; this permits the standing reduction $|c|>2$ on p. 143.

Lemma 2 (p. 143) gives the coefficient $p_n(c,q)$ of the target series explicitly in terms of two Gaussian $q$-binomial coefficients,

$$p_n(c,q)=\sum_{k=0}^{n-1}(-c)^kq^{k(k+3)/2}{n-1\brack k}_q{n+k-1\brack n-1}_q,$$

an identity obtained from the Cauchy binomial theorem. In particular, $p_n\in\mathbb Z[c,q]$ and has degree $n-1$ in $c$; the paper notes that this stronger integrality information improves the irrationality estimates but is not essential for irrationality itself. Lemma 3 (p. 143) clears the denominators arising in the residue formula: after multiplication by

$$(n-2)!\prod_{k=1}^{n}(1-cq^k)\prod_{k=\lfloor n/2\rfloor}^{n}(1-q^k),$$

the integral becomes a polynomial-coefficient linear form in $S(c,q)$, with the remaining term $s_n(c,q)\in\mathbb Z[c,q]$ of degree at most $2n$ in $c$. Lemma 4 (pp. 143–144), obtained by moving the contour through the poles $t=cq^m$, supplies, for $|q|\ge2$ and $|c|\ge2$, the quadratic-exponential estimate

$$|F_n(q)|\le 2^{n+1}/q^{3n^2/2}$$

(as printed, with $q$ rather than $|q|$ in the bound).

Lemma 5 (p. 144) proves eventual nonvanishing by expressing $F_n$ as $\sum_{m\ge n}I_m$ and using signs or alternation together with $|I_{m+1}|<|I_m|$. In the proof of Theorem 1 (p. 144), these facts yield nonzero integer linear forms tending to zero after the rational parameter $c=\alpha/\beta$ is cleared by $\beta^{2n}$, contradicting rationality.

For Theorem 2, Borwein introduces an alternating analogue $F_n^*(q)$ (p. 145). Clearing its denominators uses the multiplier $(n-2)!\prod_{k=1}^{n}(1-q^k)\prod_{k=1}^{n}(1-cq^k)\prod_{k=[n/3]}^{n}(1+q^k)$, whose last product arises from the terms at the pole at zero. This gives an integral polynomial-coefficient form

$$G_n(q)=\alpha_n(c,q)\sum_{h\ge1}\frac{(-1)^h}{1-cq^h}+\beta_n(c,q)$$

with the estimate $0<|G_n(q)|\le n!D^n/q^{n^2/18}$ (as printed) for a constant $D=D_{q,c}$ (pp. 145–146); nonvanishing is said to follow essentially as in Lemma 5.

Finally, the unnumbered concluding paragraph on p. 146 asserts that both families are not Liouville. The paper says that an asymptotic refinement of Lemma 4 and standard irrationality-measure arguments give an inequality $|\alpha-p/q|>q^{-N}$ for some fixed $N$; this stronger assertion is sketched rather than stated as a numbered theorem or proved in full detail. The introduction's Lambert-series identity $\sum_{n\ge1}(2^n-1)^{-1}=\sum_{n\ge1}d(n)2^{-n}$ and attribution of its irrationality to Erdős are cited background, not new results of the paper (p. 141).

Relation to E257

This source bears on Problem 257.

Write the E257 quantity as

$$X_A=\sum_{n\in A}\frac1{2^n-1}.$$

For $A=\mathbb N$, Theorem 1 applies directly with the theorem's parameters $q=2$ and $r=-1$, proving $X_{\mathbb N}$ irrational. Equivalently, in the proof's auxiliary notation, $X_{\mathbb N}=-S(1,2)$. Consequently, the theorem also settles every cofinite $A$: removing finitely many terms changes $X_{\mathbb N}$ by a rational number.

More generally, it settles a single infinite arithmetic progression, including any finite modification or tail of one. If

$$A=\{a+dk:k\ge0\},\qquad a,d\ge1,$$

then, after separating the rational $k=0$ term,

$$X_A=\frac1{2^a-1}+2^{-a}\sum_{k\ge1}\frac1{(2^d)^k-2^{-a}}.$$

Theorem 1 applies to the latter series with $q=2^d$ and $r=-2^{-a}$, so $X_A$ is irrational. This is a genuine E257 special case, but it does not extend merely by adding several progression sums, since irrationality of the individual summands does not exclude rational cancellation.

The potentially reusable part of the paper is its construction of nonzero, rapidly vanishing integer linear forms: equation (1) and Lemmas 1–5 (pp. 142–144) provide the model, while Lemma 2 identifies the needed $q$-binomial integrality. An argument for general $A$ would need an analogue whose coefficient of $X_A$ is integral after controlled denominator clearing and whose error remains nonzero and quadratically small.

The decisive limitation is equation (2). For the complete series, shifting $c$ to $cq^m$ removes exactly a finite initial segment. For a subseries

$$S_A(c,q)=\sum_{n\in A}\frac1{1-cq^n},$$

one instead has

$$S_A(cq^m,q)=\sum_{j\in A+m}\frac1{1-cq^j},$$

which generally differs from $S_A(c,q)$ in infinitely many terms. Thus the residue reduction and its consecutive-product arithmetic do not survive an arbitrary indicator set $A$. Theorem 2 only treats the fixed sign pattern $(-1)^n$, not arbitrary zero-one selection. Accordingly, the paper supplies important structured special cases and a possible linear-form template, but it neither states nor proves E257 for every infinite $A$.

Relation to E264

This source bears on Problem 264.

Write E264's sequence as $a_n=n!$. Borwein's input sequence is instead $b_n=q^n$, and his theorems concern only the two specially structured sums

$$\sum_{n\ge1}\frac1{b_n+r},\qquad \sum_{n\ge1}\frac{(-1)^n}{b_n+r},$$

where the same rational shift $r$ is used for every $n$. Thus Theorems 1 and 2 do not apply after setting $b_n=a_n$: their residue calculations, equation (2), the $q$-binomial formula in Lemma 2, and the denominator-clearing products in Lemma 3 all depend on the constant-ratio identity $b_{n+m}=b_nb_m$. For factorials, $a_{n+1}/a_n=n+1$, and there is no corresponding substitution in the paper.

The potentially reusable ingredient is the proof architecture. To attack a particular factorial series by this route, one would seek integer linear forms

$$L_N=A_N\xi+B_N,$$

with $A_N,B_N\in\mathbb Z$, $L_N\ne0$, and $L_N\to0$. Lemma 3 illustrates arithmetic denominator clearing, Lemma 4 gives contour-based smallness, and Lemma 5 isolates the separate nonvanishing step. None of those lemmas supplies such forms for $n!$, however, and the paper proves no irrationality statement even for the fixed-shift factorial sums $\sum 1/(n!+r)$.

Moreover, Problem 264 records that $2^n$ is not an irrationality sequence in its sense (Kovač and Tao, Corollary 2.6), whereas Borwein proves irrationality of every admissible fixed-shift sum built from $q^n$. Consequently, irrationality of these fixed-shift geometric series is strictly insufficient for the predicate occurring in Problem 264, which quantifies over every bounded integer sequence $(b_n)$ with $b_n\ne0$ rather than over a single shift. The paper is relevant mainly as a model for constructing small nonzero integral linear forms, not as a reduction or partial resolution of E264.

Relation to E1050

This source bears on Problem 1050.

Problem 1050 asks whether $\sum_{n\ge1}1/(2^n-3)$ is irrational, which Borwein settled in the 1991 paper cited here as [2]. Theorem 1 reproves that result and extends it: it gives the irrationality of $\sum_{n\ge1}1/(q^n+r)$ for every integer $q$ with $|q|>1$ and every nonzero rational $r\ne-q^m$, where [2] handles only $q>0$. The E1050 series is the case $q=2$, $r=-3$, so Theorem 1 supplies a second, self-contained proof of the problem's answer, by contour integrals in place of the Padé approximation of [2].

Results.

• Theorem 1 (p. 142): $\sum_{n\ge1}1/(q^n+c)$ is irrational for integer $|q|>1$ and nonzero rational $c\ne-q^n$.
• Theorem 2 (p. 145): $\sum_{n\ge1}(-1)^n/(q^n+c)$ is irrational under the same hypotheses.

Read status. Claims checked: the statements of Theorems 1 and 2 were read clause by clause on the printed pages; the proofs were read for structure only.

Bears on. #1050 (Theorem 1 with $q=2$, $c=-3$ is the problem's series; the theorem reproves the answer of the 1991 paper), #257 (Theorem 1 with $q=2^d$, $c=-2^{-a}$, as worked out above, gives the sum over a single arithmetic progression and, with $q=2$, $c=-1$, over every cofinite set; the paper states neither case and nothing about a general infinite set; Theorem 2 gives only a difference of two such sums), #264 (context only: a constant shift of $q^n$, neither the factorial case nor the problem's predicate for $2^n$)

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.