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arxiv: 1906.11991 · v1 · pith:L7YFL75Lnew · submitted 2019-06-27 · 🧮 math.NT

Applications of the Heine and Bauer-Muir transformations to Rogers-Ramanujan type continued fractions

Jongsil Lee , James Mc Laughlin , Jaebum Sohn This is my paper

Pith reviewed 2026-05-25 14:13 UTC · model grok-4.3

classification 🧮 math.NT
keywords continued fractionsRamanujan functionsBauer-Muir transformationHeine transformationq-seriesinfinite productsRogers-Ramanujan
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The pith

Bauer-Muir transformations relate multiple continued fractions for the ratio of general Ramanujan functions, and Heine's continued fraction produces a new one for the same ratio.

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

The paper establishes that different continued fractions expressing the quotient G(aq,b,lq)/G(a,b,l) can be obtained from one another by direct application of the Bauer-Muir transformation. It further shows these fractions arise from Heine's known continued fraction for a ratio of basic hypergeometric series, producing at least one new continued fraction for the quotient. The argument relies on separate convergence of the numerator and denominator sequences to confirm that the transformed fractions equal the original ratio. The same approach yields fresh continued-fraction forms for certain combinations of infinite products that Ramanujan had previously expressed.

Core claim

Various continued fractions for the quotient of general Ramanujan functions G(aq,b,lq)/G(a,b,l) may be derived from each other via Bauer-Muir transformations, and these continued fractions may also be derived from Heine's continued fraction, yielding a new continued fraction for G(aq,b,lq)/G(a,b,l).

What carries the argument

Bauer-Muir transformation applied to continued fractions for the quotient G(aq,b,lq)/G(a,b,l), together with Heine's continued fraction for a ratio of _2 phi_1 series.

If this is right

  • Known continued fractions for the same Ramanujan-function ratio are interconvertible by the transformation.
  • A previously unrecorded continued fraction for G(aq,b,lq)/G(a,b,l) follows from Heine's fraction.
  • New continued-fraction expansions exist for the combinations of infinite products that Ramanujan expressed in other forms.
  • The method supplies systematic ways to generate further identities among Rogers-Ramanujan-type continued fractions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same transformation technique may generate additional identities among continued fractions attached to other basic hypergeometric ratios.
  • Iterated application of Bauer-Muir steps could produce infinite families of distinct expansions for a single ratio.
  • Numerical verification of the new continued fraction for specific q-values would provide immediate checks on the claimed equality.

Load-bearing premise

The numerators and denominators of the continued fractions converge separately.

What would settle it

A concrete choice of parameters a, b, l, q where the original ratio equals one value but a Bauer-Muir transform converges to a different value because the numerator or denominator series diverges.

read the original abstract

In this paper we show that various continued fractions for the quotient of general Ramanujan functions $G(aq,b,\l q)/G(a,b,\l)$ may be derived from each other via Bauer-Muir transformations. The separate convergence of numerators and denominators play a key part in showing that the continued fractions and their Bauer-Muir transformations converge to the same limit. We also show that these continued fractions may be derived from Heine's continued fraction for a ratio of $_2\phi_1$ functions and other continued fractions of a similar type, and by this method derive a new continued fraction for $G(aq,b,\l q)/G(a,b,\l)$. Finally we derive a number of new versions of some beautiful continued fraction expansions of Ramanujan for certain combinations of infinite products, with the following being an example: \begin{multline*} \frac{(-a,b;q)_{\infty} - (a,-b;q)_{\infty}}{(-a,b;q)_{\infty}+ (a,-b;q)_{\infty}} = \frac{(a-b)}{1-a b} \- \frac{(1-a^2)(1-b^2)q}{1-a b q^2}\\ \- \frac{(a-bq^2)(b-aq^2)q}{1-a b q^4} %\phantom{sdsadadsaasdda}\\ \- \frac{(1-a^2q^2)(1-b^2q^2)q^3}{1-a b q^6} \- \frac{(a-bq^4)(b-aq^4)q^3}{1-a b q^8} \- \cds . \end{multline*}

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.

Referee Report

1 major / 2 minor

Summary. The paper claims that continued fractions for the ratio G(aq,b,lq)/G(a,b,l) of general Ramanujan functions can be inter-derived from one another via Bauer-Muir transformations, with the separate convergence of numerators and denominators ensuring that the transformed fractions share the same limit as the originals. It further shows that these fractions arise from Heine's continued fraction for a ratio of _2φ1 series (and similar forms), yielding a new continued fraction for the target ratio, and derives new continued-fraction expansions for certain combinations of infinite q-products that extend Ramanujan's classical identities.

Significance. If the convergence arguments are made explicit, the work supplies a systematic method for relating and generating Rogers-Ramanujan-type continued fractions from established q-hypergeometric transformations, together with concrete new product identities. This aligns with and extends classical techniques in q-series without introducing new ad-hoc parameters or circular appeals.

major comments (1)
  1. [Abstract (and the sections presenting the Bauer-Muir applications)] The central claim that Bauer-Muir transformations of the continued fractions for G(aq,b,lq)/G(a,b,l) converge to the same limit as the original fractions rests on the separate convergence of numerators and denominators. The abstract states this role explicitly, yet the manuscript supplies no explicit derivation of the required convergence conditions (via ratio tests on the partial numerators/denominators or asymptotic analysis of the q-Pochhammer factors) that would hold uniformly in the general parameter regime. This step is load-bearing for the inter-derivability and the derivation from Heine's fraction.
minor comments (2)
  1. [Abstract] In the displayed multline equation in the abstract, the line containing %phantom{sdsadadsaasdda} should be cleaned of the commented-out LaTeX command before publication.
  2. Notation for the general Ramanujan function G(a,b,l) and the parameter l should be introduced with a brief reminder of its definition (or a reference) at first use to aid readers unfamiliar with the specific normalization.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation for major revision. The single major comment identifies a genuine gap in the explicit justification of convergence, which we will address by adding the requested derivations.

read point-by-point responses

  1. Referee: [Abstract (and the sections presenting the Bauer-Muir applications)] The central claim that Bauer-Muir transformations of the continued fractions for G(aq,b,lq)/G(a,b,l) converge to the same limit as the original fractions rests on the separate convergence of numerators and denominators. The abstract states this role explicitly, yet the manuscript supplies no explicit derivation of the required convergence conditions (via ratio tests on the partial numerators/denominators or asymptotic analysis of the q-Pochhammer factors) that would hold uniformly in the general parameter regime. This step is load-bearing for the inter-derivability and the derivation from Heine's fraction.

    Authors: We agree that the manuscript invokes the separate convergence of numerators and denominators without supplying the explicit ratio-test or asymptotic arguments that would hold uniformly across the general parameter regime. This omission weakens the load-bearing step. We will insert a dedicated subsection (or short appendix) that derives the necessary convergence conditions for the partial numerators and denominators, using ratio tests on the q-Pochhammer factors and standard estimates for |q|<1. The revised text will also clarify how these conditions guarantee that the Bauer-Muir transforms share the same limit as the original continued fractions. revision: yes

Circularity Check

0 steps flagged

No circularity; derivations apply known transformations to standard identities

full rationale

The paper starts from the established Heine continued fraction for _2φ1 ratios and the Bauer-Muir transformation, then applies them to quotients of general Ramanujan G-functions using q-Pochhammer symbol identities. The separate-convergence assumption is invoked to equate limits but is not derived from the target results themselves; it functions as an external hypothesis rather than a self-referential step. No equations reduce by construction to fitted parameters, renamed known results, or self-citation chains that carry the central claims. The new continued fraction for G(aq,b,lq)/G(a,b,l) is obtained by explicit transformation steps from Heine's fraction, keeping the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivations rest on classical properties of q-Pochhammer symbols, basic hypergeometric series, and convergence criteria for continued fractions; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • standard math Standard algebraic and convergence properties of q-Pochhammer symbols and _2phi_1 series
    Used to define the Ramanujan functions and the starting Heine continued fraction.
  • domain assumption Separate convergence of numerators and denominators implies the transformed continued fraction has the same limit
    Invoked to justify that Bauer-Muir transformations preserve the value.

pith-pipeline@v0.9.0 · 5853 in / 1369 out tokens · 51458 ms · 2026-05-25T14:13:43.336626+00:00 · methodology

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Reference graph

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