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arxiv: 1906.11997 · v1 · pith:FNBTJFJYnew · submitted 2019-06-27 · 🧮 math.NT

Mock Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series

James Mc Laughlin This is my paper

Pith reviewed 2026-05-25 14:10 UTC · model grok-4.3

classification 🧮 math.NT
keywords mock theta functionsbilateral seriesbasic hypergeometric seriesBailey transformationssummation formulasroots of unitytheta functions
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The pith

Many third-, fifth-, sixth- and eighth-order mock theta functions arise exactly as special cases of the bilateral 2ψ2 series.

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

The paper establishes that bilateral series versions of many third-, fifth-, sixth- and eighth-order mock theta functions can be obtained directly as special cases of the general 2ψ2 series. Applying three transformation formulas for this series due to Bailey produces new transformation and summation identities for both the mock theta functions and their bilateral series. Summation formulas are further used to show that, for a mock theta function χ(q) and suitable roots of unity ζ, the difference χ(q) minus an associated theta function θ_χ(q) possesses a finite limit as q approaches ζ from inside the unit disk.

Core claim

The bilateral series corresponding to many of the third-, fifth-, sixth- and eighth order mock theta functions may be derived as special cases of 2ψ2 series. Three transformation formulae for this series due to Bailey are used to derive various transformation and summation formulae for both these mock theta functions and the corresponding bilateral series. New and existing summation formulae for these bilateral series are also used to make explicit in a number of cases the fact that for a mock theta function, say χ(q), and a root of unity in a certain class, say ζ, that there is a theta function θ_χ(q) such that the limit as q→ζ of (χ(q)−θ_χ(q)) exists, as q→ζ from within the unit circle.

What carries the argument

The bilateral basic hypergeometric 2ψ2 series together with Bailey's three transformation formulae applied to it.

If this is right

  • New transformation formulas hold for the mock theta functions via the 2ψ2 identities.
  • New summation formulas hold for the associated bilateral series.
  • Explicit finite limits exist for χ(q) minus θ_χ(q) at certain roots of unity.
  • The same summation formulas apply to produce further identities for both mock theta functions and bilateral series.

Where Pith is reading between the lines

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

  • The method could extend to mock theta functions of additional orders once matching special cases of 2ψ2 are identified.
  • The root-of-unity limit results may connect to known asymptotic behaviors of mock theta functions in the theory of partitions.
  • Further summation formulas not derived here might follow from additional known identities for the 2ψ2 series.

Load-bearing premise

The listed mock theta functions match exactly the indicated special cases of the bilateral 2ψ2 series without extra convergence or analytic continuation conditions that would block direct use of Bailey's transformations.

What would settle it

A direct term-by-term comparison or numerical check for a generic |q|<1 value showing that at least one listed mock theta function fails to equal the corresponding 2ψ2 special case.

read the original abstract

The bilateral series corresponding to many of the third-, fifth-, sixth- and eighth order mock theta functions may be derived as special cases of $_2\psi_2$ series \[ \sum_{n=-\infty}^{\infty}\frac{(a,c;q)_n}{(b,d;q)_n}z^n. \] Three transformation formulae for this series due to Bailey are used to derive various transformation and summation formulae for both these mock theta functions and the corresponding bilateral series. \\ New and existing summation formulae for these bilateral series are also used to make explicit in a number of cases the fact that for a mock theta function, say $\chi(q)$, and a root of unity in a certain class, say $\zeta$, that there is a theta function $\theta_{\chi}(q)$ such that \[ \lim_{q \to \zeta}(\chi(q) - \theta_{\chi}(q)) \] exists, as $q \to \zeta$ from within the unit circle.

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 / 1 minor

Summary. The paper shows that bilateral series for many third-, fifth-, sixth- and eighth-order mock theta functions arise as special cases of the _2ψ2 bilateral basic hypergeometric series ∑_{n=-∞}^∞ (a,c;q)_n / (b,d;q)_n z^n. Three transformation formulae for this series due to Bailey are applied to obtain transformation and summation formulae for the mock theta functions and the bilateral series. New and existing summation formulae are then used to prove that, for a mock theta function χ(q) and certain roots of unity ζ, the limit as q→ζ (from inside the unit disk) of χ(q) − θ_χ(q) exists, where θ_χ is a theta function.

Significance. If the derivations are valid, the work supplies a uniform hypergeometric framework for generating identities among mock theta functions and their bilateral companions, extending Bailey’s transformations to this setting. The explicit construction of the limiting theta-function differences at roots of unity furnishes concrete analytic information that may be useful for further study of mock theta functions inside the unit disk.

major comments (2)
  1. [Abstract and §2 (specializations)] The central claim requires that each listed mock theta function equals a specific specialization of the _2ψ2 series (e.g., a=0, c=q, z=q or analogous choices) and that Bailey’s three transformations may be invoked verbatim. The standard convergence region for the bilateral _2ψ2 is the annulus |q|<|z|<1; the mock-theta specializations place parameters at 0 or 1, producing zero or infinite Pochhammer symbols and locating the series on the boundary. The manuscript must supply an explicit verification that these parameter choices lie inside the domain where Bailey’s identities hold without additional analytic continuation.
  2. [§4 (limiting statements)] The later use of summation formulae to establish existence of lim_{q→ζ}(χ(q)−θ_χ(q)) presupposes that the transformed series remains valid under this limiting process. No explicit justification is indicated that the transformed bilateral series converges or can be continued to the required roots of unity; this step is load-bearing for the second main claim.
minor comments (1)
  1. Define all q-Pochhammer notation and bilateral-series conventions at the first appearance rather than assuming familiarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the specific comments on analytic justification. We address each major point below and indicate where revisions will be made to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract and §2 (specializations)] The central claim requires that each listed mock theta function equals a specific specialization of the _2ψ2 series (e.g., a=0, c=q, z=q or analogous choices) and that Bailey’s three transformations may be invoked verbatim. The standard convergence region for the bilateral _2ψ2 is the annulus |q|<|z|<1; the mock-theta specializations place parameters at 0 or 1, producing zero or infinite Pochhammer symbols and locating the series on the boundary. The manuscript must supply an explicit verification that these parameter choices lie inside the domain where Bailey’s identities hold without additional analytic continuation.

    Authors: The specializations are chosen so that the bilateral series reduce precisely to the defining q-series of the mock theta functions inside |q|<1, where those series converge. Bailey’s transformations are identities valid in the open annulus; the boundary cases are obtained by taking appropriate limits of the parameters (e.g., a→0) while keeping |q|<1, which is justified because both sides of each transformed identity remain holomorphic in that disk. Nevertheless, we agree that an explicit paragraph is desirable. In the revised manuscript we will insert, at the end of §2, a short verification for each family of specializations, confirming that the indeterminate Pochhammer symbols are resolved by the standard limiting definitions and that the resulting identities hold by continuity within |q|<1. revision: yes

  2. Referee: [§4 (limiting statements)] The later use of summation formulae to establish existence of lim_{q→ζ}(χ(q)−θ_χ(q)) presupposes that the transformed series remains valid under this limiting process. No explicit justification is indicated that the transformed bilateral series converges or can be continued to the required roots of unity; this step is load-bearing for the second main claim.

    Authors: After the Bailey transformations the mock theta function minus the accompanying theta function is expressed as a bilateral series whose summands become, at q=ζ, either zero or terms that combine into a convergent theta series. Because the original mock theta is holomorphic for |q|<1 and the transformed expression equals it inside the disk, the limit as q→ζ from inside the disk exists by continuity. We concede that the manuscript does not spell out this interchange explicitly. In the revision we will add one sentence in §4 noting that the transformed bilateral series converges absolutely in a punctured neighborhood of each admissible root of unity inside the unit disk, permitting passage to the limit term by term or by identification with a known theta function. revision: yes

Circularity Check

0 steps flagged

No circularity; derivations apply external Bailey transformations to defined special cases of _2ψ2 series.

full rationale

The paper's chain consists of identifying mock theta functions as special cases of the bilateral _2ψ2 series and then invoking three pre-existing transformation formulae due to Bailey (an external reference) to obtain new identities and summation results. The abstract and described claims contain no self-definitional steps, no parameters fitted to data then relabeled as predictions, and no load-bearing self-citations. The limit statements are presented as direct consequences of the applied summation formulae rather than reductions to the paper's own inputs by construction. The work is therefore self-contained against external benchmarks (Bailey's transformations) and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard properties of q-Pochhammer symbols, convergence of bilateral basic hypergeometric series, and the validity of Bailey's three transformation formulae; no free parameters or invented entities are introduced.

axioms (2)
  • standard math Standard algebraic and analytic properties of the q-Pochhammer symbol (a;q)_n and bilateral basic hypergeometric series
    Invoked throughout the identification of mock theta bilateral series as special cases of _2ψ_2 and in the application of transformations.
  • standard math Bailey's three transformation formulae for the _2ψ_2 series hold under the stated parameter conditions
    Used as the engine to derive the new identities for mock theta functions.

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Reference graph

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