arxiv: 2604.09726 · v1 · submitted 2026-04-09 · 🧮 math.GM

Recognition: 2 theorem links

· Lean Theorem

Error terms for continued fractions of e^{1/s} and sqrt{frac{v}{u}}tanh\!Bigl(frac{1}{sqrt{uv}}Bigr)

Nikita Kalinin , Takao Komatsu

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Pith reviewed 2026-05-10 17:54 UTC · model grok-4.3

classification 🧮 math.GM

keywords continued fractionserror termstelescoping sumsexponential functionhyperbolic tangentconvergentsapproximation

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The pith

Error terms in the continued fractions for e to the 1/s and sqrt(v/u) tanh(1/sqrt(uv)) satisfy weighted sum identities that recover the target values and the values plus one.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper starts from the observation that any real number alpha given by a continued fraction admits two telescoping decompositions: the partial quotients a_n weighted by the absolute errors of the convergents sum to alpha plus one, while the same quotients weighted by the squared errors sum exactly to alpha. It then inserts the known explicit continued fraction for e to the power 1/s, whose partial quotients follow the repeating three-term block that grows linearly, and the corresponding expansion for the scaled hyperbolic tangent, and records that the same two sum identities hold verbatim for these targets. A reader would care because the identities turn the sequence of rational approximations into an exact additive decomposition of the transcendental function itself.

Core claim

For the continued fraction e^{1/s} = [1; repeating (2k-1)s-1, 1, 1] and the continued fraction for sqrt(v/u) tanh(1/sqrt(uv)), the general decompositions yield sum a_{n+1} |E_n| = alpha + 1 and sum a_{n+1} E_n^2 = alpha, where E_n = p_n - alpha q_n are the signed errors of the convergents p_n/q_n.

What carries the argument

The signed error E_n = p_n - alpha q_n of each convergent p_n/q_n, which satisfies a linear recurrence with the partial quotients a_n that permits exact telescoping of the weighted sums.

Load-bearing premise

The general telescoping identities for arbitrary continued fractions apply without modification to these two specific infinite expansions whose partial quotients grow linearly with the index.

What would settle it

Compute the first several hundred convergents of the continued fraction for e^{1/1} from the given partial quotients, form the partial sums of a_{n+1} |E_n| and a_{n+1} E_n^2, and check whether they approach e plus one and e respectively.

read the original abstract

Many classical identities arise from nothing more mysterious than looking at the same object in two different ways. A number, a function, or a combinatorial object may admit several natural decompositions, and by disassembling it in one way and reassembling it in another, we often obtain unexpected corollaries. Telescoping sums provide a particularly vivid incarnation of this principle: by arranging terms so that successive contributions cancel, one performs a conceptual ``cut-and-paste'' that often admits a clean geometric interpretation. Generating functions offer a complementary perspective. Encoding a problem into a formal power series and then evaluating that series at a prescribed point naturally expresses the same quantity as an infinite (or finite) expansion, and equating these representations yields a wealth of identities. For example, for a real number $\alpha$ given by its continued fraction expansion $\alpha = [a_0, a_1,a_2,\dots]$, with convergents $p_n/q_n$ and error terms $E_n := p_n - \alpha q_n$, one can obtain ``additive'' decompositions of the form $\sum_{n\ge-1} a_{n+1}\,\lvert E_n\rvert \;=\; \alpha + 1$, $\sum_{n\ge-1} a_{n+1}\,E_n^{2} \;=\; \alpha$. Thus $\alpha$ and $\alpha+1$ themselves appear as weighted sums of the local approximation errors of their convergents. In this note we explore what such decompositions yield in two explicit cases: the continued fraction \[ e^{1/s} = [1;\,{\overline{(2k-1)s-1,1,1}}]_{k=1}^{\infty} \] and the continued fraction \[ \frac{s}{u}\tanh\!\Bigl(\frac{1}{s}\Bigr) = [\,0;\,\overline{(4k-3)u,\,(4k-1)\tfrac{s^{2}}{u}}\,]_{k=1}^{\infty} \]

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This note applies general telescoping error-sum identities for continued fractions to the known expansions of e^{1/s} and a scaled tanh, producing explicit weighted sums that follow from the usual recurrences without modification.

read the letter

This paper takes two known continued fraction expansions—one for e to the power 1/s with a repeating pattern of partial quotients, and one for a hyperbolic tangent expression—and applies general telescoping sum formulas for the approximation errors to them. The result is a pair of identities: the sum of a_{n+1} times the absolute error equals the number plus one, and the sum of a_{n+1} times the squared error equals the number itself. The authors frame it as exploring what the general decompositions produce in these cases. What stands out is how little extra machinery is needed. The general formulas follow directly from the convergent recurrences, the determinant relation for consecutive convergents, and the fact that the errors vanish at infinity. None of that depends on the specific form of the partial quotients, so the linear growth in these examples just makes the series converge faster but doesn't require any adjustment. The central claims hold up because the underlying relations are standard and unrestricted. On the downside, the work stays at the level of application. There are no new proofs of the general identities, no numerical verifications of the sums for specific parameter values, and no discussion of how these might connect to other properties of e or tanh. If the full text is mostly the abstract expanded with the two cases written out, it feels like a short note rather than a substantial development. This kind of paper is for people who work with continued fractions of transcendental functions and like seeing explicit identities fall out of general principles. A reader already familiar with the error sum formulas would find it a quick confirmation for these two cases. It is not going to change how anyone thinks about continued fractions broadly, but it does deliver concrete results cleanly. I would mention it in a reading group focused on special functions or Diophantine approximation if time allows, but it is not essential. I would not cite it in my own work unless I needed exactly these sums. Still, it is solid enough and formally grounded that a serious editor should send it to peer review rather than desk reject it. The math checks out, and the scope is modest but honest.

Referee Report

0 major / 3 minor

Summary. The manuscript presents general telescoping identities for any continued fraction α = [a_0; a_1, ...] with convergents p_n/q_n and errors E_n = p_n - α q_n, namely ∑_{n≥-1} a_{n+1} |E_n| = α + 1 and ∑_{n≥-1} a_{n+1} E_n² = α. It then applies these identities to the explicit continued fraction e^{1/s} = [1; repeating (2k-1)s-1, 1, 1] and to the continued fraction (s/u) tanh(1/s) = [0; repeating (4k-3)u, (4k-1)s²/u], deriving the corresponding weighted error sums for these functions.

Significance. The general identities follow directly from the standard recurrences p_n = a_n p_{n-1} + p_{n-2}, q_n = a_n q_{n-1} + q_{n-2}, the determinant relation, and E_n → 0, making them parameter-free and broadly applicable; this is a clear strength. Specializing to the classical continued fractions for e^{1/s} and the indicated tanh expression produces concrete sum representations that may facilitate new analyses or verifications in approximation theory and special functions. The work aligns with its stated theme of obtaining identities via multiple decompositions of the same object.

minor comments (3)

[Abstract] Abstract: the second continued fraction is presented as (s/u) tanh(1/s), while the title uses the more general form √(v/u) tanh(1/√(uv)); the manuscript should explicitly relate the two or indicate whether the title describes a parameterized generalization.
[General identities section] The sum indexing n ≥ -1 (with implied E_{-1} = 1) should be defined explicitly at the outset, as standard continued-fraction notation often begins indexing at n = 0.
[General identities section] A brief reference to a standard source (e.g., Khinchin, Continued Fractions) for the determinant identity E_n q_{n-1} - E_{n-1} q_n = (-1)^{n-1} would aid readers unfamiliar with the background.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and encouraging report. The summary accurately captures the main contributions of the general telescoping identities and their specializations to the continued fractions for e^{1/s} and the indicated tanh expression. We are pleased that the parameter-free nature of the identities and their potential utility in approximation theory are highlighted as strengths.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The general identities sum a_{n+1} |E_n| = α + 1 and sum a_{n+1} E_n² = α are derived from the standard continued-fraction recurrences p_n = a_n p_{n-1} + p_{n-2}, q_n = a_n q_{n-1} + q_{n-2} together with the determinant relation E_n q_{n-1} - E_{n-1} q_n = (-1)^{n-1} and the limit E_n → 0. These relations are valid for any convergent continued fraction and do not depend on the particular linear growth of the partial quotients in the two explicit expansions. The paper simply substitutes the known continued-fraction representations of e^{1/s} and (s/u) tanh(1/s) into the already-established general formulas; no step reduces the target sums to a fitted parameter, a self-citation, or a redefinition of the input quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are mentioned; the work rests on standard properties of continued fractions and convergents.

pith-pipeline@v0.9.0 · 5710 in / 1149 out tokens · 44221 ms · 2026-05-10T17:54:17.489285+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

for a real number α given by its continued fraction expansion α=[a0;a1,…], … one can obtain additive decompositions of the form ∑ a_{n+1} |E_n| = α+1, ∑ a_{n+1} E_n² = α
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

e^{1/s} = [1; repeating (2k-1)s-1,1,1] and √(v/u) tanh(1/√(uv))

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

21 extracted references · 1 canonical work pages

[1]

Elsner, On error sums of square roots of positive integers with applications to Lucas and Pell numbers,J

C. Elsner, On error sums of square roots of positive integers with applications to Lucas and Pell numbers,J. Integer Seq.17(2014), no. 4, Article 14.4.4, 21 pp

2014
[2]

Elsner and M

C. Elsner and M. Stein, On error sum functions formed by convergents of real numbers,J. Integer Seq.14(2011), no. 8, Article 11.8.6, 14 pp

2011
[3]

Euler, De fractionibus continuis dissertatio,Commentarii academiae scien- tiarum Petropolitanae9(1744), 98–137

L. Euler, De fractionibus continuis dissertatio,Commentarii academiae scien- tiarum Petropolitanae9(1744), 98–137
[4]

Hermite, Sur la fonction exponentielle,C

Ch. Hermite, Sur la fonction exponentielle,C. R. Acad. Sci. Paris77(1873)
[5]

Hetyei, Hurwitzian continued fractions containing a repeated constant and an arithmetic progression,SIAM J

G. Hetyei, Hurwitzian continued fractions containing a repeated constant and an arithmetic progression,SIAM J. Discrete Math.28(2014), no. 2, 962–985

2014
[6]

Hurwitz, ¨Uber die Entwicklung complexer Gr¨ oßen in Kettenbr¨ uche,Acta Math.11(1887–1888), 187–200

A. Hurwitz, ¨Uber die Entwicklung complexer Gr¨ oßen in Kettenbr¨ uche,Acta Math.11(1887–1888), 187–200
[7]

Kalinin, Legendre duality for certain summations over the Farey pairs, Preprint, arXiv:2409.10592v4, 2025

N. Kalinin, Legendre duality for certain summations over the Farey pairs, Preprint, arXiv:2409.10592v4, 2025

work page arXiv 2025
[8]

Kalinin and M

N. Kalinin and M. Shkolnikov, Tropical formulae for summation over a part of SL(2,Z),Eur. J. Math.5(2019), no. 3, 909–928

2019
[9]

Komatsu, A proof of the continued fraction expansion ofe 2/s,Integers7 (2009), no

T. Komatsu, A proof of the continued fraction expansion ofe 2/s,Integers7 (2009), no. 1, Paper A30, 8 p

2009
[10]

Komatsu, A Diophantine approximation ofe 1/s in terms of integrals,Tokyo J

T. Komatsu, A Diophantine approximation ofe 1/s in terms of integrals,Tokyo J. Math.32(2009), no. 1, 159–176

2009
[11]

Komatsu, Diophantine approximations of tanh, tan, and linear forms of ein terms of integrals,Rev

T. Komatsu, Diophantine approximations of tanh, tan, and linear forms of ein terms of integrals,Rev. Roumaine Math. Pures Appl.54(2009), no. 3, 223–242

2009
[12]

Komatsu, Some exact algebraic expressions for the tails of Tasoev continued fractions,J

T. Komatsu, Some exact algebraic expressions for the tails of Tasoev continued fractions,J. Aust. Math. Soc.92(2012), no. 2, 179–193

2012
[13]

Lambert, M´ emoire sur quelques propri´ et´ es remarquables des quantit´ es tran- scendentes circulaires et logarithmiques, 1768

M. Lambert, M´ emoire sur quelques propri´ et´ es remarquables des quantit´ es tran- scendentes circulaires et logarithmiques, 1768. In:π: A Source Book, Springer, New York, 2004, 129–140

2004
[14]

Mc Laughlin, Some new families of Tasoevian and Hurwitzian continued fractions,Acta Arith.135(2008), no

J. Mc Laughlin, Some new families of Tasoevian and Hurwitzian continued fractions,Acta Arith.135(2008), no. 3, 247–268

2008
[15]

D. N. Lehmer, Arithmetical theory of certain Hurwitzian continued fractions, Amer. J. Math.40(1918), no. 4, 375–390

1918
[16]

K. R. Matthews and R. F. C. Walters, Some properties of the continued frac- tion expansion of (m/n)e 1/q,Proc. Cambridge Philos. Soc.67(1970), 67–74

1970
[17]

C. D. Olds, The simple continued fraction expansion ofe,Amer. Math. Monthly 77(1970), no. 9, 968–974

1970
[18]

T. J. Osler, A proof of the continued fraction expansion ofe 1/M,Amer. Math. Monthly113(2006), no. 1, 62–66. ERROR TERMS FOR CONTINUED FRACTIONS OFe 1/s AND p v u tanh 1√uv 19

2006
[19]

A. J. van der Poorten, Continued fraction expansions of values of the expo- nential function and related fun with continued fractions,Nieuw Arch. Wisk. (4)14(1996), no. 2, 221–230

1996
[20]

J. N. Ridley and G. Petruska, The error-sum function of continued fractions, Indag. Math.(N.S.)11(2000), no. 2, 273–282

2000
[21]

B. G. Tasoev, On rational approximations of some numbers,Mat. Zametki67 (2000), no. 6, 931–937; translation inMath. Notes67(2000), no. 5–6, 786–791. Mathematics and Computer Science Department, Guangdong Tech- nion Israel Institute of Technology, Shantou 515603, China Email address:nikita.kalinin@gtiit.edu.cn Institute of Mathematics, Henan Academy of Sci...

2000