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arxiv: 2510.17209 · v5 · submitted 2025-10-20 · 🧮 math.CO

On Bilateral Multiple Sums and Rogers-Ramanujan Type Identities

Dandan Chen , Tianjian Xu This is my paper

Pith reviewed 2026-05-18 06:42 UTC · model grok-4.3

classification 🧮 math.CO MSC 05A1711P8433D15
keywords Rogers-Ramanujan identitiesbilateral sumsbasic hypergeometric seriesmulti-sum identitiesq-seriespartition identities
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The pith

New bilateral double-sum Rogers-Ramanujan identities with parameters are proven and then applied to generate multi-sum versions.

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

The paper proves several new identities that express bilateral double sums in closed form using the style of Rogers-Ramanujan identities but with additional parameters included. These proofs combine the theory of basic hypergeometric series with an integral evaluation technique. The resulting identities are then applied directly to produce several new multi-sum Rogers-Ramanujan type identities. A reader would care because such identities expand the known catalog of q-series equalities that arise in partition theory and related areas.

Core claim

We establish some new bilateral double-sum Rogers-Ramanujan identities involving parameters. As applications, these identities yield several new multi-sum Rogers-Ramanujan type identities. Our proofs utilize the theory of basic hypergeometric series in conjunction with the integral method.

What carries the argument

Basic hypergeometric series theory combined with the integral method for closing bilateral multiple sums.

If this is right

  • Several new multi-sum Rogers-Ramanujan type identities follow by direct application of the bilateral double-sum cases.
  • The identities hold in parameterized form, allowing specialization to recover or extend earlier results.
  • The same proof technique supplies closed forms for the double sums that serve as the starting point for the multi-sum derivations.

Where Pith is reading between the lines

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

  • The parameterized bilateral forms may admit further specializations that connect to existing single-sum or triple-sum identities in the literature.
  • The integral-plus-hypergeometric approach could be tested on other families of bilateral sums to produce analogous closed forms.
  • Special cases of the new identities might admit combinatorial interpretations in terms of restricted partitions or lattice paths.

Load-bearing premise

The integral method combined with basic hypergeometric series theory produces valid closed forms for the bilateral sums without unstated restrictions on parameter ranges or convergence.

What would settle it

Direct numerical evaluation of both sides of one claimed identity for concrete parameter values where the series converge absolutely.

read the original abstract

We establish some new bilateral double-sum Rogers-Ramanujan identities involving parameters. As applications, these identities yield several new multi-sum Rogers-Ramanujan type identities. Our proofs utilize the theory of basic hypergeometric series in conjunction with the integral method.

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 manuscript establishes new bilateral double-sum Rogers-Ramanujan identities involving parameters, proved via the theory of basic hypergeometric series combined with an integral method. As applications, the authors derive several new multi-sum Rogers-Ramanujan type identities.

Significance. If the identities and their derivations hold with appropriate parameter restrictions, the work adds to the literature on q-series and partition identities by providing new bilateral and multi-sum examples. The combination of hypergeometric techniques with integral representations is a recognized approach that can yield closed forms not easily obtained by other means.

major comments (2)
  1. [§3] §3 (main bilateral identities): the integral representations for the bilateral double sums are applied to obtain closed forms, but no explicit conditions on the parameters (such as |q|<1 together with Re(a)>0 or |a|<1 for the relevant q-Pochhammer symbols) are stated to guarantee absolute convergence and justify interchange of sum and integral. This is load-bearing because bilateral sums from −∞ to ∞ are sensitive to pole locations.
  2. [§4] §4 (applications to multi-sum identities): the derivations of the new multi-sum Rogers-Ramanujan type identities inherit the same unstated convergence restrictions from the bilateral sums; without verified domains, the claimed identities hold only conditionally rather than for the full parameter ranges asserted.
minor comments (2)
  1. [§2] Notation for the bilateral sums could be clarified by explicitly indicating the summation indices and the precise form of the summands in the first displayed equation of §2.
  2. A brief comparison table or list of previously known bilateral Rogers-Ramanujan identities would help situate the new results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and for emphasizing the need for explicit convergence conditions. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3] §3 (main bilateral identities): the integral representations for the bilateral double sums are applied to obtain closed forms, but no explicit conditions on the parameters (such as |q|<1 together with Re(a)>0 or |a|<1 for the relevant q-Pochhammer symbols) are stated to guarantee absolute convergence and justify interchange of sum and integral. This is load-bearing because bilateral sums from −∞ to ∞ are sensitive to pole locations.

    Authors: We agree that the manuscript should include explicit parameter restrictions to ensure absolute convergence of the bilateral sums and to justify the interchange of summation and integration. In the revised version, we will add a paragraph at the beginning of Section 3 stating the required conditions, such as |q| < 1 together with suitable restrictions on the parameters (e.g., Re(a) > 0 or |a| < 1 where appropriate for the q-Pochhammer symbols involved) to guarantee the validity of the integral representations. revision: yes

  2. Referee: [§4] §4 (applications to multi-sum identities): the derivations of the new multi-sum Rogers-Ramanujan type identities inherit the same unstated convergence restrictions from the bilateral sums; without verified domains, the claimed identities hold only conditionally rather than for the full parameter ranges asserted.

    Authors: We acknowledge that the multi-sum identities in Section 4 are derived from the bilateral identities of Section 3 and therefore inherit the same convergence requirements. In the revision, we will explicitly reference the convergence conditions established in Section 3 when stating the multi-sum identities and will restrict the asserted parameter ranges accordingly so that the claims are precise. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard q-series methods applied to new cases

full rationale

The paper derives new bilateral double-sum Rogers-Ramanujan identities by applying the established theory of basic hypergeometric series together with the integral method. These techniques rely on known summation formulas and integral representations that are independent of the specific parameter choices and new identities presented. No steps reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the proofs are self-contained against external benchmarks in q-series literature.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the established theory of basic hypergeometric series and the applicability of an integral method; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • standard math The theory of basic hypergeometric series applies directly to the bilateral sums under consideration.
    Invoked as the primary proof tool.
  • domain assumption The integral method yields valid transformations or evaluations for these series.
    Combined with hypergeometric theory to obtain the identities.

pith-pipeline@v0.9.0 · 5547 in / 1224 out tokens · 28919 ms · 2026-05-18T06:42:29.221183+00:00 · methodology

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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