Theta functions and transformations of bilateral basic hypergeometric series

Nian Hong Zhou

arxiv: 2605.05094 · v2 · submitted 2026-05-06 · 🧮 math.NT · math.CA· math.CO

Theta functions and transformations of bilateral basic hypergeometric series

Nian Hong Zhou This is my paper

Pith reviewed 2026-05-15 06:10 UTC · model grok-4.3

classification 🧮 math.NT math.CAmath.CO

keywords theta functionsbilateral basic hypergeometric seriesq-seriesasymptotic expansionstransformation formulasMcIntosh conjecturesRamanujan identities

0 comments

The pith

New transformation formulas connect theta functions to bilateral basic hypergeometric series and prove McIntosh conjectures on q-series asymptotics.

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

The paper derives new transformation formulas that relate theta functions to certain bilateral basic hypergeometric series. These formulas are used to construct companion q-series for a given class of q-series such that the quotient of the two admits an asymptotic expansion with a simple closed form. The resulting identities establish several conjectures of McIntosh on asymptotic transformations of q-series. They also extend earlier identities due to Ramanujan and McIntosh. A reader would care because the work supplies explicit tools for controlling the long-term growth of q-series that appear throughout number theory and special functions.

Core claim

The central claim is that new transformation formulas involving theta functions and bilateral basic hypergeometric series permit the construction of companion q-series for which the asymptotic expansion of the quotient takes a simple closed form. This directly proves several conjectures of McIntosh on asymptotic transformations of q-series and extends identities previously found by Ramanujan and McIntosh.

What carries the argument

Transformation formulas that incorporate theta functions into bilateral basic hypergeometric series, enabling the construction of companion q-series whose quotients have simplified asymptotic expansions.

Load-bearing premise

The transformations remain valid and the asymptotic expansions of the quotients simplify precisely on the parameter domains where the bilateral series converge and the relevant expansions exist.

What would settle it

A concrete set of parameters inside the convergence region of the bilateral series where the constructed quotient's asymptotic expansion fails to reduce to the claimed closed form would disprove the transformations.

read the original abstract

We establish new transformation formulas involving theta functions and certain bilateral basic hypergeometric series. From these formulas, we construct companion $q$-series for a class of $q$-series such that the asymptotic expansion of their quotient admits a simple closed form. This allows us to prove several conjectures of McIntosh on asymptotic transformations of $q$-series. Moreover, our results extend some identities of Ramanujan and McIntosh.

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.

Desk Editor's Note private letter to a colleague

New theta-based transformations for bilateral q-series that settle several McIntosh conjectures, with the main gap being explicit convergence domains.

read the letter

The paper's core contribution is a set of new transformation identities that link theta functions to certain bilateral basic hypergeometric series. These identities are then used to build companion series whose quotients have simple closed-form asymptotic expansions, which in turn prove several of McIntosh's open conjectures on q-series behavior. The work also recovers and extends a few known Ramanujan-type identities as special cases. That is the actual advance: concrete formulas that turn previously conjectural asymptotics into theorems inside the theory of basic hypergeometric series.

Referee Report

1 major / 0 minor

Summary. The paper establishes new transformation formulas involving theta functions and bilateral basic hypergeometric series _rψ_s. From these, it constructs companion q-series for a class of q-series such that the asymptotic expansion of their quotient admits a simple closed form. This is used to prove several conjectures of McIntosh on asymptotic transformations of q-series and to extend some identities of Ramanujan and McIntosh.

Significance. If the transformations are established with full rigor, the work would supply useful tools for asymptotic analysis of q-series and bilateral hypergeometric functions, resolve specific open conjectures of McIntosh, and provide extensions of classical identities. The combination of theta-function transformations with companion-series constructions for quotients is a potentially fruitful approach in the area.

major comments (1)

[Main transformation theorems] The statements of the new transformation formulas (appearing after the abstract and in the main results section) do not explicitly delimit the parameter domains on which the bilateral series converge (typically |q|<1 and z in an annulus determined by the parameters) nor the conditions under which the constructed quotients admit the claimed asymptotic expansions. This omission is load-bearing because the proofs of the McIntosh conjectures rely on equating the asymptotic behavior of the quotients; without these domains and any required analytic-continuation arguments, the conjectures are only formally verified rather than rigorously established for the stated range.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for explicit convergence domains. We agree that these must be stated clearly to ensure the proofs are fully rigorous rather than formal, and we will revise the manuscript accordingly.

read point-by-point responses

Referee: The statements of the new transformation formulas (appearing after the abstract and in the main results section) do not explicitly delimit the parameter domains on which the bilateral series converge (typically |q|<1 and z in an annulus determined by the parameters) nor the conditions under which the constructed quotients admit the claimed asymptotic expansions. This omission is load-bearing because the proofs of the McIntosh conjectures rely on equating the asymptotic behavior of the quotients; without these domains and any required analytic-continuation arguments, the conjectures are only formally verified rather than rigorously established for the stated range.

Authors: We agree with this assessment. In the revised version we will add explicit statements of the convergence regions for each bilateral series _rψ_s (requiring |q|<1 together with z lying in an annulus |q|^α < |z| < |q|^β whose exponents α, β are determined by the parameters). We will likewise specify the parameter restrictions under which the companion-series quotients possess the claimed asymptotic expansions, including any necessary analytic-continuation steps that justify equating the asymptotic behaviors. These additions will render the proofs of the McIntosh conjectures rigorous over the stated ranges. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivations rest on independent transformations

full rationale

The paper claims to establish new transformation formulas for bilateral basic hypergeometric series using theta functions, then applies them to construct companion q-series and prove McIntosh conjectures while extending Ramanujan-McIntosh identities. No quoted step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain. The central results are presented as fresh identities valid on convergence domains, with no evidence that any prediction or uniqueness claim loops back to the paper's own inputs or prior self-work. The skeptic concern addresses domain delimitation and analytic continuation, which is a rigor issue rather than circularity. This is the expected self-contained case for a q-series identity paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; full paper may contain additional convergence axioms or parameter restrictions not visible here.

axioms (2)

domain assumption Bilateral basic hypergeometric series converge in specified parameter domains
Standard prerequisite for the series to be well-defined before transformations can be applied.
domain assumption Asymptotic expansions of the constructed quotients admit simple closed forms
The central construction relies on this property holding after the theta-function transformations.

pith-pipeline@v0.9.0 · 5350 in / 1401 out tokens · 49247 ms · 2026-05-15T06:10:53.258904+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We establish new transformation formulas involving theta functions and certain bilateral basic hypergeometric series... L_α(x;q,z) = ∑_{n∈Z} z^n q^{α(n/2)} / (x)_n
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_strictMono_of_one_lt unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Corollary 1.1 ... asymptotic formula with exp(α t/8 + α/(2t)(log z)^2) and O(e^{-2π²/(α t)})

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

[1]

G. E. Andrews.The theory of partitions. Encyclopedia of Mathematics and its Applications, Vol. 2. Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976

work page 1976
[2]

G. E. Andrews.Partitions: yesterday and today. New Zealand Mathematical Society, Wellington, 1979. With a foreword by J. C. Turner

work page 1979
[3]

G. E. Andrews and B. C. Berndt.Ramanujan’s lost notebook. Part II. Springer, New York, 2009

work page 2009
[4]

Dixit and G

A. Dixit and G. Kumar. The Rogers-Ramanujan dissection of a theta function.Math. Ann., 393(1):667– 696, 2025

work page 2025
[5]

Gasper and M

G. Gasper and M. Rahman.Basic hypergeometric series, volume 96 ofEncyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, second edition, 2004. With a foreword by Richard Askey

work page 2004
[6]

Lewin.Polylogarithms and associated functions

L. Lewin.Polylogarithms and associated functions. North-Holland Publishing Co., New York- Amsterdam, 1981. With a foreword by A. J. Van der Poorten

work page 1981
[7]

R. J. McIntosh. Some asymptotic formulae forq-hypergeometric series.J. London Math. Soc. (2), 51(1):120–136, 1995

work page 1995
[8]

R. J. McIntosh. Asymptotic transformations ofq-series.Canad. J. Math., 50(2):412–425, 1998

work page 1998
[9]

M. J. Schlosser. Bilateral identities of the Rogers-Ramanujan type.Trans. Amer. Math. Soc. Ser. B, 10:1119–1140, 2023

work page 2023
[10]

G. N. Watson. The Final Problem : An Account of the Mock Theta Functions.J. London Math. Soc., S1-11(1):55. N. H. Zhou: School of Mathematics and Statistics, The Center for Applied Mathematics of Guangxi, Guangxi Normal University, Guilin 541004, Guangxi, PR China Email address:nianhongzhou@outlook.com; nianhongzhou@gxnu.edu.cn 19

work page

[1] [1]

G. E. Andrews.The theory of partitions. Encyclopedia of Mathematics and its Applications, Vol. 2. Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976

work page 1976

[2] [2]

G. E. Andrews.Partitions: yesterday and today. New Zealand Mathematical Society, Wellington, 1979. With a foreword by J. C. Turner

work page 1979

[3] [3]

G. E. Andrews and B. C. Berndt.Ramanujan’s lost notebook. Part II. Springer, New York, 2009

work page 2009

[4] [4]

Dixit and G

A. Dixit and G. Kumar. The Rogers-Ramanujan dissection of a theta function.Math. Ann., 393(1):667– 696, 2025

work page 2025

[5] [5]

Gasper and M

G. Gasper and M. Rahman.Basic hypergeometric series, volume 96 ofEncyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, second edition, 2004. With a foreword by Richard Askey

work page 2004

[6] [6]

Lewin.Polylogarithms and associated functions

L. Lewin.Polylogarithms and associated functions. North-Holland Publishing Co., New York- Amsterdam, 1981. With a foreword by A. J. Van der Poorten

work page 1981

[7] [7]

R. J. McIntosh. Some asymptotic formulae forq-hypergeometric series.J. London Math. Soc. (2), 51(1):120–136, 1995

work page 1995

[8] [8]

R. J. McIntosh. Asymptotic transformations ofq-series.Canad. J. Math., 50(2):412–425, 1998

work page 1998

[9] [9]

M. J. Schlosser. Bilateral identities of the Rogers-Ramanujan type.Trans. Amer. Math. Soc. Ser. B, 10:1119–1140, 2023

work page 2023

[10] [10]

G. N. Watson. The Final Problem : An Account of the Mock Theta Functions.J. London Math. Soc., S1-11(1):55. N. H. Zhou: School of Mathematics and Statistics, The Center for Applied Mathematics of Guangxi, Guangxi Normal University, Guilin 541004, Guangxi, PR China Email address:nianhongzhou@outlook.com; nianhongzhou@gxnu.edu.cn 19

work page