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Continued Fractions of Polynomial Type: Theory and Encyclopedic Dictionary

T0 review · 2 major / 4 minor · reviewed 2026-07-12 · grok-4.5

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Pith's one-line read Continued fractions of polynomial type admit a complete generic classification of exact convergence speeds, which the paper uses to build a large dictionary of expansions for constants and special functions, many new and all annotated by ra

desk verdict Huge usable dictionary of polynomial-type CFs with rates and systematic Apéry acceleration; theory is openly heuristic/generic and many identities are unproved guesses. read the letter →

arxiv 2607.06581 v1 pith:C3YAVESN submitted 2026-07-03 math.NT

Henri Cohen This is my paper · ORCID
classification math.NT MSC 11A5511J70
keywords continuedfractionspolynomialtypeconvergenceratesBauer-MuiraccelerationApérymethodspecialconstantszetavalueshypergeometricseries
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies continued fractions whose coefficients eventually become rational functions of the index (or of even and odd indices), called of polynomial type. It supplies a generic asymptotic classification of how fast the convergents approach their limit, expressed in a compact factorial-exponential-subexponential-polynomial-logarithmic taxonomy determined solely by the degrees and leading coefficients of the numerator and denominator sequences. It then develops systematic acceleration via the classical Bauer-Muir transformation iterated in the manner introduced by Apéry, which frequently converts polynomially slow fractions into exponentially convergent ones of period two. The resulting theory is applied to produce an encyclopedic dictionary containing more than sixteen hundred explicit expansions for classical constants (powers of two, pi, logarithms, zeta values, Catalan's constant, etc.) and for many special functions, each accompanied by its precise speed of convergence (sometimes only up to a multiplicative constant). Most of these expansions appear to be new, and almost none previously carried rigorous rate information. A companion computational package makes the constructions and numerical checks routine.

What carries the argument

Theorem 2.1 (the FEDP classification of asymptotic types from the expansions a(n) ~ a0 n^alpha and b(n) ~ b0 n^beta) together with the Bauer-Muir-Apéry acceleration process that produces an infinite family of new equivalent or Möbius-related fractions of strictly higher speed, typically of period two.

What would settle it

Compute several hundred convergents of any claimed new accelerated fraction (for example an Apéry-accelerated expansion for 2^{1/3} or for zeta(3)) to high precision and verify whether the observed absolute error matches both the predicted limit and the exact FEDP rate; a systematic mismatch for generic initial data would falsify the corresponding entry or the classification itself.

Watch

Extended reading notes

Core claim

For any convergent continued fraction of polynomial type the error after n steps is generically of the form C over (n!^F E^n exp(sqrt(D n)) n^P), where the four parameters F, E, D, P are completely determined by comparing the degrees alpha and beta of a(n) and b(n) and by the signs and magnitudes of their leading coefficients; the possible regimes are factorial, exponential, two sub-exponential types, two polynomial types, logarithmic, and separate even/odd polynomial limits. These rates can be improved, often dramatically, by iterated Bauer-Muir transformations followed by Apéry's diagonal (or dual) extraction, yielding new fractions whose coefficients remain of polynomial type.

Load-bearing premise

The speed classification holds only generically and rests on a partly heuristic asymptotic analysis of the ratio of successive denominators; many of the explicit dictionary entries themselves were obtained by pattern-guessing from initial terms rather than by rigorous closed-form proofs.

Editorial extensions

If this is right

  • High-precision numerical evaluation of many classical constants becomes possible with fully quantified error bounds coming from the classified rates.
  • Apéry-style acceleration applies routinely far beyond the original zeta(2) and zeta(3) cases, producing exponentially convergent expansions for large classes of periods and gamma quotients.
  • The dictionary supplies ready-made starting points for Diophantine-approximation arguments that require explicit rapidly convergent continued fractions.
  • The accompanying computational tools allow systematic generation, contraction, and numerical validation of further families of polynomial-type expansions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same degree-based classification and acceleration pipeline is likely to extend, with only minor changes, to continued fractions whose coefficients grow like polynomials times exponential or q-Pochhammer factors.
  • Most of the series representations listed beside the dictionary entries are hypergeometric and therefore candidates for fully rigorous creative-telescoping or Wilf-Zeilberger proofs.
  • The fastest expansions given for algebraic numbers such as cube roots already imply effective irrationality measures once the multiplicative constants C are rigorously bounded.
  • Logarithmic-type fractions can often be converted into polynomially convergent ones by a single Bauer-Muir step, after which ordinary extrapolation becomes practical.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The manuscript develops the theory of continued fractions of polynomial type (period T with rational-function coefficients for large n) and supplies an encyclopedic dictionary of more than 1600 explicit examples for constants and special functions. Chapters 1–4 treat basic transformations (Euler series/product maps, equivalence, contractions), a multi-case asymptotic classification of convergence speed (Theorem 2.1: types F, E, D±, P±, L, Pnc expressed in FEDP notation), numerical evaluation and extrapolation, and Bauer–Muir–Apéry acceleration (including duals, multipliers, simplifiers and “slowed” variants). Chapters 5–6 list the fractions themselves, each entry giving the closed form (as a Pari/GP closure), the coefficient vectors, a few terms, the claimed type and rate (often only up to a multiplicative constant C), a short asymptotic expansion A, and frequently a series representation. Appendices cover asymptotic expansions, implementation details, hypergeometric identities and the accompanying Pari/GP database. The author repeatedly flags that Theorem 2.1 is generic/heuristic and that many dictionary identities were guessed “à la Ramanujan” from initial terms and remain unproved.

Significance. If the classification and the bulk of the dictionary entries hold, the work fills a genuine gap: the literature almost never records precise convergence rates, and the systematic application of iterated Bauer–Muir–Apéry acceleration produces many new, rapidly convergent expansions for classical constants (π, log 2, ζ(3), Catalan’s constant, 2^{1/3}, …) and for special functions. The accompanying machine-readable Pari/GP database and the explicit acceleration algorithms are concrete, reusable contributions that go beyond a mere compilation. Even under the stated genericity caveats the catalogue is likely to become a standard reference for numerical and Diophantine applications of polynomial-type continued fractions.

major comments (2)
  1. Theorem 2.1 (Chapter 2) and the accompanying sketch in §2.3 are the sole engine that attaches a precise rate to every dictionary entry. The proof is explicitly labelled “heuristic” and “generic” (Remarks 2.2); it assumes an asymptotic expansion of the ratio v(n)=q(n)/q(n-1) of a form that is not justified for all initial data. Because the same engine is used for the rates claimed throughout Chapters 5–6, any non-generic failure invalidates those rates. A rigorous justification (or a clear delimitation of the exceptional set) is therefore load-bearing for the central claim of the dictionary.
  2. Introduction to Chapter 5 states that many closed forms were obtained by guessing from initial terms and that the author “does not claim” proofs. While numerical verification via cflimit/cfasymp is reported, the absence of even a sketch of proof (or a systematic verification protocol) for a representative sample of the “probably new” entries leaves open the possibility that some identities are simply false. Given that the dictionary is presented as the main product, this verification gap is material.
minor comments (4)
  1. The FEDP notation [F,E,D,P] is introduced in §2.2 but is used with slight variations (sometimes omitting C, sometimes writing only the type letter). A single consistent definition early in Chapter 2 would help.
  2. Several parametric families (e.g., 5.2.4, 5.3.52) list “complicated conditions” on the parameters without an explicit arithmetic description; a short table or a reference to the generating script would improve usability.
  3. Typos and sign errors in the asymptotic constants C are acknowledged by the author; a systematic re-check of a random sample of the C-values (especially those involving square roots of quadratic irrationals) would reduce the expected error rate.
  4. The Pari/GP package is currently available only on private GIT branches. A stable DOI or permanent archive link for both the package and the dictionary file would make the computational claims reproducible.

Circularity Check

0 steps flagged · score 1.0 of 10

No load-bearing circularity: convergence types and dictionary rates rest on a stated heuristic asymptotic ansatz for v(n), not on redefining targets as their CFs or on self-citation that forces the claims.

full rationale

The paper's central technical claim is Theorem 2.1 classifying the speed of polynomial-type CFs into types F/E/D±/P±/L/Pnc from the degrees and leading coefficients of a(n) and b(n). The proof sketch (Section 2.3) reduces to a(n)=1, sets v(n)=q(n)/q(n-1), and assumes an asymptotic power series for v(n); the author repeatedly labels the result generic and heuristic (Remarks 2.2). That is a methodological limitation (correctness risk), not circularity: the claimed rates are not obtained by fitting the target constant into the CF or by renaming a known identity. Euler transforms (Prop. 1.7 and corollaries) and Bauer–Muir–Apéry recursions (Chapter 4, Prop. 4.1) are independent algebraic constructions that produce new CFs from series or from prior CFs; their limits are not forced by definition of the input. Dictionary entries that are Euler transforms of known series, or Apéry accelerations of those, inherit their values from the series; entries obtained 'à la Ramanujan' by guessing (Chapter 5 introduction) introduce verification risk, not circular definition. Self-citation of the author's Pari/GP package is tooling for numerical checks and asymptotics, not a uniqueness theorem that forbids alternatives. No step reduces a claimed prediction to a fitted parameter or to a self-citation chain that equates claim to input. Score 1 for the minor self-reference to the author's software and prior books as implementation context only.

Assumptions & free parameters 0 free parameters · 4 assumptions · 2 invented entities

The work rests on classical CF algebra, a definition of polynomial type, and a heuristic asymptotic analysis of three-term recurrences. No free parameters are fitted to physical data. The main non-standard load is the genericity/heuristic status of the convergence taxonomy and the unproved status of many dictionary formulas.

assumptions (4)
  • standard math Standard three-term recurrence and determinant identities for convergents p(n)/q(n) of a continued fraction (a(n),b(n)).
    Chapter 1; used throughout for equivalence, contractions, and error formulas.
  • ad hoc to paper Asymptotics of q(n) and of S−p(n)/q(n) for polynomial-type CFs are given by the multi-case classification in Theorem 2.1 under generic initial data and heuristic expansions of v(n)=q(n)/q(n−1).
    Chapter 2 explicitly labels the result generic and the proof partly heuristic; this underpins almost all stated speeds.
  • domain assumption Bauer–Muir and Apéry diagonal/vertical constructions preserve the limit up to a Möbius transformation with small integer coefficients when r(n) is chosen so that d(n) is suitably simple.
    Chapter 4 and Appendix B; standard in the literature but applied here at large scale, with initial terms often fixed a posteriori via fracdep.
  • ad hoc to paper Many dictionary closed forms equal the stated constants/functions for suitable ranges of parameters, even when only guessed from initial terms.
    Chapter 5 introduction: author does not claim proofs for many entries.
invented entities (2)
  • Continued fractions of polynomial type (period T with rational-function coefficients for large n) independent evidence
    purpose: Restrict the class so that convergence taxonomy and acceleration stay algebraic and implementable.
    Definition 1.5; a classification device rather than a physical entity; independent evidence is the classical theory of rational CFs.
  • FEDP notation [F,E,D,P] for convergence type independent evidence
    purpose: Compactly report asymptotic error shapes across the dictionary.
    Chapter 2 notation; pure bookkeeping, not a new mathematical object with independent ontology.

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Cite this review

Pith. "Pith review of Continued Fractions of Polynomial Type: Theory and Encyclopedic Dictionary." pith.science (2026). https://pith.science/paper/C3YAVESN

@misc{pith2026260706581,
  author       = {Pith},
  title        = {Pith review of: Continued Fractions of Polynomial Type: Theory and Encyclopedic Dictionary},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C3YAVESN}},
  note         = {Machine review of arXiv:2607.06581}
}
read the original abstract

After giving a number of properties of continued fractions of polynomial type, in particular focusing on convergence properties and Bauer-Muir-Ap\'ery acceleration techniques, we give a large list of continued fractions, both for specific real numbers, and for special functions, some extracted from a number of different sources, but most others being probably new. In addition to providing such a list, one of our main additions is to include the exact speed of convergence of these continued fractions (sometimes only up to a multiplicative constant), which is almost always omitted in the literature.

Discussion (0). Continue with ORCID to comment.

Reference graph

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