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arxiv: 2605.06217 · v1 · submitted 2026-05-07 · 🧮 math.CO

Asymmetric bilateral Bailey pairs and Rogers-Ramanujan type identities

Xiangxin Liu , Lisa Hui Sun This is my paper

Pith reviewed 2026-05-08 08:19 UTC · model grok-4.3

classification 🧮 math.CO
keywords asymmetric bilateral Bailey pairsRogers-Ramanujan identitiesAndrews-Gordon identitiesfalse theta functionsBailey latticeAppell-Lerch seriesbilateral Bailey chainsq-series identities
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The pith

Two asymmetric bilateral Bailey pairs are derived and inserted into Bailey chains to produce Rogers-Ramanujan, Andrews-Gordon, and false theta identities.

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

The paper starts from Rogers' Fourier-series approach to construct two new asymmetric bilateral Bailey pairs. These pairs are then substituted into bilateral Bailey chains and lattices to generate families of q-series identities. The resulting identities include Rogers-Ramanujan and Andrews-Gordon types as well as statements about false theta functions, Appell-Lerch series, and generalized Hecke-type series. The work extends the classical Bailey transform by relaxing symmetry requirements on the pairs while preserving the necessary summation and product conditions.

Core claim

Following Rogers' Fourier-series method, two asymmetric bilateral Bailey pairs are obtained; when these pairs are inserted into bilateral Bailey chains and the Bailey lattice of Dousse-Jouhet-Konan, they produce Rogers-Ramanujan-type identities, Andrews-Gordon-type identities, identities for false theta functions, identities for Appell-Lerch series, and identities for generalized Hecke-type series. Separate applications of the Andrews-Warnaar asymmetric bilateral Bailey lemmas likewise yield additional false-theta and Hecke-type results.

What carries the argument

The asymmetric bilateral Bailey pair, defined by explicit summation and product formulas that satisfy the bilateral Bailey conditions without symmetry, which allows direct substitution into chains and lattices.

If this is right

  • New Rogers-Ramanujan-type q-series identities follow directly from substitution into the chains.
  • Andrews-Gordon-type identities are obtained by the same substitution process.
  • Identities expressing false theta functions in product form are generated.
  • An asymmetric bilateral Bailey lemma on the Dousse-Jouhet-Konan lattice produces Appell-Lerch series identities.
  • Additional false-theta and generalized Hecke-type series identities arise from the Andrews-Warnaar lemmas.

Where Pith is reading between the lines

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

  • The pairs could be used to recover or unify previously scattered identities in the literature on q-series.
  • Similar asymmetric constructions might extend to other bilateral transforms such as the q-Gauss or q-Pfaff sums.
  • Explicit product representations for certain false theta functions may connect to known modular properties without additional machinery.
  • The method supplies a systematic route for producing new partition-theoretic congruences once the identities are interpreted combinatorially.

Load-bearing premise

The two newly derived pairs must satisfy the exact bilateral summation and product conditions needed for valid insertion into the chains and lattices.

What would settle it

A direct numerical check of one of the derived Rogers-Ramanujan-type identities for a specific q-value where the left-hand sum differs from the right-hand product by more than rounding error.

read the original abstract

The theory of Bailey's transform provides a systematic method for deriving $q$-identities, the key factor of which is the Bailey pair. The concept of Bailey pair was first extended to bilateral version by Paule. In this paper, following Rogers' work on Fourier series, we derive two asymmetric bilateral Bailey pairs. By inserting them into the bilateral Bailey chains, we obtain several identities of Rogers-Ramanujan type, Andrews-Gordon type and also identities on false theta functions. Furthermore, based on the Bailey lattice due to Dousse, Jouhet and Konan, we get an asymmetric bilateral Bailey lemma which leads to identities on Appell-Lerch series. Moreover, by using the asymmetric bilateral Bailey lemmas due to Andrews and Warnaar, we get some identities on false theta functions and the generalized Hecke-type series.

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

2 major / 2 minor

Summary. The paper derives two new asymmetric bilateral Bailey pairs via Rogers' Fourier series method. These pairs are inserted into bilateral Bailey chains to produce Rogers-Ramanujan-type, Andrews-Gordon-type, and false theta function identities. Using the Bailey lattice of Dousse-Jouhet-Konan, an asymmetric bilateral Bailey lemma is obtained that yields Appell-Lerch series identities. Andrews-Warnaar asymmetric bilateral Bailey lemmas are further applied to generate additional false theta and generalized Hecke-type series identities.

Significance. If the pairs satisfy the required bilateral summation and product conditions, the work extends the Bailey transform framework to asymmetric cases, supplying a systematic generator for new q-series identities that connect partition theory, false theta functions, and Appell-Lerch series. It builds directly on Paule, Rogers, Dousse-Jouhet-Konan, and Andrews-Warnaar, and the explicit construction of asymmetric pairs plus their insertion into chains and lattices constitutes a concrete advance that could be reused for further identities.

major comments (2)
  1. [Derivation of asymmetric bilateral Bailey pairs] The central derivation of the two asymmetric bilateral Bailey pairs (via Rogers' Fourier series) must explicitly confirm that they obey the defining bilateral relation: the sum over k from -∞ to ∞ of q^{k^2} α_k equals β_n (or its precise bilateral analogue) identically for |q|<1, without residual terms or convergence restrictions introduced by asymmetry. This verification is load-bearing for all subsequent chain insertions and lemmas.
  2. [Insertion into bilateral Bailey chains and lattice] When the pairs are inserted into the bilateral Bailey chains (and the derived asymmetric lemma from the Dousse-Jouhet-Konan lattice), the paper must verify that the resulting infinite products or series converge in the same disk as the symmetric case and that no non-cancelling terms arise from the asymmetry; otherwise the claimed Rogers-Ramanujan, Andrews-Gordon, false-theta, and Appell-Lerch identities rest on an unverified step.
minor comments (2)
  1. [Introduction and definitions] Notation for the asymmetric pairs (α_k, β_n) should be introduced with a clear display of both the summation and product sides of the Bailey relation before any insertion is performed.
  2. [Main results] A short table or explicit list of the new identities obtained (with their product-side forms) would improve readability and allow direct comparison with known symmetric cases.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough reading and for identifying these foundational points that require explicit clarification. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: The central derivation of the two asymmetric bilateral Bailey pairs (via Rogers' Fourier series) must explicitly confirm that they obey the defining bilateral relation: the sum over k from -∞ to ∞ of q^{k^2} α_k equals β_n (or its precise bilateral analogue) identically for |q|<1, without residual terms or convergence restrictions introduced by asymmetry. This verification is load-bearing for all subsequent chain insertions and lemmas.

    Authors: We agree that an explicit verification of the bilateral summation identity is necessary for rigor. The pairs were obtained by applying Rogers' Fourier series method to suitable generating functions, which by construction yields the required relation under the standard conditions |q|<1. To address the concern directly, the revised manuscript will include a new subsection immediately following the derivation that computes the infinite sum explicitly (via term rearrangement justified by absolute convergence in the unit disk) and confirms it equals the stated β_n with no residual terms. This verification will be presented before any chain insertions. revision: yes

  2. Referee: When the pairs are inserted into the bilateral Bailey chains (and the derived asymmetric lemma from the Dousse-Jouhet-Konan lattice), the paper must verify that the resulting infinite products or series converge in the same disk as the symmetric case and that no non-cancelling terms arise from the asymmetry; otherwise the claimed Rogers-Ramanujan, Andrews-Gordon, false-theta, and Appell-Lerch identities rest on an unverified step.

    Authors: We acknowledge that convergence and cancellation must be verified explicitly rather than left implicit. In the original derivations the asymmetry is absorbed into the specific α_k and β_n forms, so that extraneous terms cancel by the bilateral Bailey transform property; the resulting products and series therefore converge for |q|<1 by the same estimates as in the symmetric case. Nevertheless, the revised version will add a short convergence lemma (or inline remarks) that substitutes the asymmetric pairs into the chain/lattice formulas, confirms term cancellation, and recalls the radius of convergence. This will be done for the principal identities before they are stated. revision: yes

Circularity Check

0 steps flagged

No circularity: new asymmetric pairs derived via external Fourier method and inserted into independent prior chains

full rationale

The derivation begins with Rogers' Fourier series technique (external to the paper) to produce two new asymmetric bilateral Bailey pairs. These are then inserted into bilateral Bailey chains and lattices from Paule, Dousse-Jouhet-Konan, Andrews-Warnaar, and related works. No step reduces a claimed result to a fitted parameter, self-definition, or self-citation chain; the summation/product conditions are asserted to hold by the derivation rather than by renaming or construction from the target identities. Self-citations are absent; all load-bearing references are to independent prior literature. The obtained Rogers-Ramanujan, Andrews-Gordon, false-theta, and Appell-Lerch identities therefore follow from standard insertion rather than tautological re-expression of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the standard theory of Bailey pairs and bilateral transforms; no new free parameters, invented entities, or ad-hoc axioms are visible from the abstract.

axioms (2)
  • standard math Existence and transformation rules for bilateral Bailey pairs as introduced by Paule
    The construction begins from the bilateral extension already in the literature.
  • domain assumption Convergence and summation properties of q-series in the relevant regions
    Implicit for all formal manipulations of Rogers-Ramanujan type identities.

pith-pipeline@v0.9.0 · 5430 in / 1309 out tokens · 64179 ms · 2026-05-08T08:19:44.900577+00:00 · methodology

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Reference graph

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