Borwein: On the irrationality of Σ (1/(q^n+r))

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theorem_4: For every integer q greater than one and every nonzero rational r different from each q^n with n at least one, the sum over n of one over q^n minus r is irrational.

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The paper prints "Copyright © 1991 by Academic Press, Inc. All rights of reproduction in any form reserved." at the foot of its first page (printed p. 253; the OCR layer reads the copyright sign as "9") and "© 1991 Academic Press, Inc." after its abstract, every other right reserved.

Peter B. Borwein, "On the irrationality of Σ (1/(q^n+r))," Journal of Number Theory, 37(3), 253-259, 1991. https://doi.org/10.1016/s0022-314x(05)80041-1

Overview

Borwein proves irrationality for a family of shifted geometric reciprocal sums. In the paper’s sign convention, Theorem 4 (stated on p. 257, proved on pp. 257–258) states that, for an integer $q>1$ and a nonzero rational $r$ with $r\ne q^m$ for every $m\ge1$,

$$\sum_{n=1}^{\infty}\frac1{q^n-r}$$

is irrational. Equivalently, replacing $r$ by $-r$, $\sum_{n\ge1}(q^n+r)^{-1}$ is irrational for nonzero rational $r\ne-q^m$. The case $r=0$ is necessarily excluded, since $\sum_{n\ge1}q^{-n}=1/(q-1)$. The introduction (pp. 253–254) situates this as an extension of Erdős’s cited result for $\sum(2^n-1)^{-1}$; the prior results mentioned there are background, not proved in this article. The introduction (p. 253) also records that Erdős and Graham called the irrationality of $\sum(2^n-3)^{-1}$ unresolved and notes that it is a special case: Theorem 4 with $q=2$ and $r=3$ makes it irrational, and it is the series of Problem 1050.

The analytic object is the $q$-logarithm

$$L_q^*(x)=\sum_{m=1}^{\infty}\frac{x}{q^m-x}=\sum_{m=1}^{\infty}\frac{x^m}{q^m-1},$$

with the respective domain qualifications stated in (1) (p. 254). It satisfies the dilation identity $L_q^*(qx)=L_q^*(x)+x/(1-x)$, (3) (p. 254), and degenerates after normalization to $-\log(1-x)$ as $q\downarrow1$, (4) (p. 255). The diagonal Padé approximants $P_n/Q_n$ are defined by $Q_nL_q^*-P_n=O(x^{2n+1})$, (2) (p. 254).

Theorem 1(a) (pp. 255–256) gives the explicit denominator

$$Q_n(x)=q^{n^2}\sum_{i=0}^n {\binom ni}_q^2\frac{\prod_{k=0}^{i-1}(1-xq^k)}{q^{i(2n-i)}},$$

(8), and asserts that it is of degree $n$ in $x$, degree $n^2$ in $q$, and has integral coefficients. Theorem 1(b) (p. 256) supplies the corresponding arithmetic denominator control for $P_n$: it states that $P_n$ has degree $n$ in $x$ and that $P_n$ times a factor printed as the sum $\sum_{k=[n/2]}^{n}(1+q^k)$ is a polynomial in $q$ with integer coefficients. The Gaussian coefficients used here are defined through the $q$-factorial in (5)–(7) (p. 255). Theorem 2 (p. 256) gives a three-term recurrence for the normalized denominator $\overline Q_n=Q_n/q^{n^2}$, defined in (9); from it the paper deduces, for $x\in[-1,1]$, the bound $|Q_n(x)|\le C_q q^{n^2}$, (10) (p. 256). The paper says that Theorems 1 and 2 are discussed and derived in reference [5], except that the argument for Theorem 1(b) is sketched here.

The approximation input is Theorem 3 (p. 257): for $q>1$ and $0<|x|\le1$,

$$0<\left|L_q^*(x)-\frac{P_n(x)}{Q_n(x)}\right|<\frac{d_q|x|^{2n}}{q^{n(n+1)}}.$$

The strict lower bound supplies nonvanishing as well as smallness. This theorem is explicitly attributed to and proved in reference [4], rather than reproved here.

For Theorem 4, Borwein takes $x=r/q^N$ and re-indexes the first series in (1) to write $L_q^*(r/q^N)=L_q^*(r)-\sum_{n=1}^N r/(q^n-r)$ (p. 257). Multiplication by

$$T_N=\prod_{n=1}^N(q^n-r)\prod_{n=\lfloor N/2\rfloor}^N(1-q^n)$$

clears the displayed finite-sum and Padé denominators; the estimate $0<|T_N|\le e_{r,q}q^{7N(N+1)/8}$ appears on p. 258. Combining this with (10) and Theorem 3 produces polynomials $S_N(r),U_N(r)$ with integral coefficients and

$$0<|S_N(r)L_q^*(r)-U_N(r)|\le g_{r,q}\frac{|r|^{2N}}{q^{N(N+1)/8}}.$$

Writing $r=h/j$ and multiplying by $j^{2N}$ yields nonzero integer linear forms tending to zero (p. 258), which contradicts rationality of $L_q^*(r)$. Since $L_q^*(r)=r\sum_{n\ge1}(q^n-r)^{-1}$ and $r\ne0$, this proves the stated sum irrational.

Finally, the unnumbered remark on p. 258 says that the same estimates yield a finite irrationality measure: $|L_q^*(r)-s/t|>t^{-\alpha}$ for a constant $\alpha$, with $\alpha=26/3$ admissible for sufficiently large $t$. The detailed standard argument is not supplied but is referred to §11.3 of reference [3]. Thus the intended conclusion is that the value $L_q^*(r)$ is not Liouville; the printed sentence saying that “$r$ is not a Liouville number” must be read in context as referring to this value.

Relation to E264

This source bears on Problem 264.

Write the E264 candidate as $a_n=n!$, and write Borwein’s comparison sequence as $b_n=q^n$. The exact transferable statement is

$$\sum_{n\ge1}\frac1{b_n-r}\notin\mathbb Q\qquad(q\in\mathbb Z,\ q>1;\ r\in\mathbb Q\setminus\{0,q,q^2,\ldots\}),$$

by Theorem 4 (pp. 257–258). This is a theorem about every fixed admissible rational translate of one geometric sequence. It does not establish the sequence-level property asked for in E264, and any additional quantifiers implicit in “irrationality sequence” are absent from the paper.

The potentially useful contribution is an irrationality-proof template. For a sequence $a=(a_n)$ one may introduce, at least formally,

$$F_a(x)=\sum_{n\ge1}\frac{x}{a_n-x},\qquad F_a(r)=r\sum_{n\ge1}\frac1{a_n-r}.$$

Borwein’s argument succeeds for $a_n=q^n$ because it simultaneously provides: explicit Padé approximants ((2) and Theorem 1), arithmetic control of their coefficients (Theorem 1(b)), a recurrence yielding denominator growth control (Theorem 2 and (10)), a nonzero error of sufficiently rapid decay (Theorem 3), and a clearing factor $T_N$ that converts the approximation into nonzero integer linear forms (Theorem 4, pp. 257–258). An analogous package for $F_{n!}(x)=\sum_{n\ge1}x/(n!-x)$ could enter an E264 argument at precisely the integer-linear-form step.

The paper does not provide that package for factorials. Its key dilation law (3) depends on $q^{n+1}=q q^n$, while $(n+1)!=(n+1)n!$ has no fixed dilation parameter. Likewise, the explicit $Q_n$ in (8) and its recurrence in Theorem 2 are built from Gaussian binomial coefficients and a fixed $q$. Although the paper notes $[n]_q!\to n!$ in (6) (p. 255), taking $q\downarrow1$ does not yield the E264 problem: the sequence in the summand remains $q^n$, not $[n]_q!$, and the decisive error factor $q^{-n(n+1)}$ in Theorem 3 loses its decay in that limit. Thus the relation to E264 is methodological and comparative, not a reduction or partial resolution.

Results.

• Theorem 4 (p. 257): $\sum_{n\ge1}1/(q^n-r)$ is irrational for integer $q>1$ and nonzero rational $r\ne q^n$ $(n\ge1)$.

Read status. Claims checked: the statement of Theorem 4 was read clause by clause on the printed page; the proof was read for structure only. Theorems 1–3 are recorded above as the paper states them, as inputs it quotes from its references [4] and [5]; their proofs are not in this paper, apart from a short argument for Theorem 1(b).

Bears on. #1050 (Theorem 4 with $q=2$, $r=3$ is the problem's series, which the introduction names as a special case), #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.