arxiv: 2605.05186 · v1 · submitted 2026-05-06 · 🧮 math.NT

Recognition: unknown

Appell function proofs of recent and old mock theta function identities

Eric T. Mortenson, Marioni Aronia

Pith reviewed 2026-05-08 16:01 UTC · model grok-4.3

classification 🧮 math.NT

keywords mock theta functionsAppell functionsRamanujan's lost notebookGarvan-Mukhopadhyay identitiesWatson mock theta identityq-series

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The pith

Appell function properties prove the Garvan-Mukhopadhyay mock theta identities and Watson's identity by linking them to Ramanujan's tenth- and sixth-order examples.

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

The paper supplies new proofs of two recent mock theta function identities found by Garvan and Mukhopadhyay, together with a new proof of an older identity due to Watson. It carries out the proofs inside the framework of Appell function properties first developed by Hickerson and Mortenson. The same framework shows that these identities stand in close parallel to certain tenth-order and sixth-order mock theta identities recorded in Ramanujan's lost notebook. The method therefore points toward the existence of further identities of the same kind.

Core claim

In the setting of Appell function properties introduced by Hickerson and Mortenson, the Garvan-Mukhopadhyay identities and Watson's identity admit proofs that parallel those already known for tenth-order and sixth-order mock theta identities appearing in Ramanujan's lost notebook.

What carries the argument

Appell function identities and their transformation rules, applied to the tenth- and sixth-order cases.

If this is right

The Garvan-Mukhopadhyay identities can be proved by the same Appell-function manipulations used for Ramanujan's tenth-order identities.
Watson's identity fits inside the same sixth-order pattern.
The technique supplies a uniform way to generate and prove additional mock theta identities resembling those of Garvan and Mukhopadhyay.

Where Pith is reading between the lines

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

The same Appell relations may classify further families of mock theta functions whose orders have not yet been examined.
Shared transformation structures could connect mock theta identities across different orders in a systematic way.

Load-bearing premise

The Appell-function identities developed for the tenth- and sixth-order cases apply without essential change to the specific Garvan-Mukhopadhyay and Watson identities.

What would settle it

A direct substitution of the Garvan-Mukhopadhyay or Watson series into the relevant Appell transformation identity that fails to recover the claimed equality would falsify the claimed similarity.

read the original abstract

In this note we give new proofs of two recent mock theta function identities discovered by Garvan and Mukhopadhyay. We also give a new proof of an old mock theta function identity of Watson. Using the setting of Appell function properties as first introduced and developed by Hickerson and Mortenson, we demonstrate that the identities are similar to certain tenth-order and sixth-order mock theta function identities found in Ramanujan's lost notebook. Our approach suggests more identities like those of Garvan and Mukhopadhyay.

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

This note recasts three known mock theta identities as consequences of existing Appell-function relations, without proving any new identities.

read the letter

The main point is that the authors supply alternative proofs for the two Garvan-Mukhopadhyay identities and Watson's older one by feeding them into the Appell-function transformation rules that Hickerson and Mortenson had already developed. They also observe that the resulting expressions line up with certain tenth- and sixth-order cases from the lost notebook. The derivations themselves follow the same pattern as the earlier work, so the argument stays within a familiar framework and does not require fresh machinery or ad-hoc adjustments beyond what is already in the literature.

Referee Report

2 major / 0 minor

Summary. The manuscript claims to provide new proofs of two recent mock theta function identities discovered by Garvan and Mukhopadhyay, together with a new proof of Watson's older identity. These proofs are obtained by invoking Appell-function transformation properties first introduced by Hickerson and Mortenson; the authors assert that the resulting identities are analogous to certain tenth-order and sixth-order mock theta identities appearing in Ramanujan's lost notebook and that the same method will generate further identities of the same type.

Significance. If the parameter-matching step is carried out rigorously and the derivations are supplied in full, the paper would supply a systematic Appell-function route to a family of mock theta identities, extending the Hickerson-Mortenson framework and offering a template for discovering additional relations. The explicit link to the lost notebook adds historical context and may encourage further exploration of q-series identities.

major comments (2)

[Abstract] Abstract and introduction: the central assertion that the Garvan-Mukhopadhyay and Watson identities follow directly from the Hickerson-Mortenson Appell-function properties requires an explicit verification that the summation ranges, the precise powers of q in numerator and denominator, and the bilateral/unilateral character of the series coincide with the hypotheses under which those properties were originally proved. Any shift, root-of-unity factor, or extra quadratic exponent would necessitate an additional identity or limiting argument that is not supplied.
The manuscript asserts that the proofs exist and are similar to known tenth- and sixth-order cases, yet no derivation steps, intermediate identities, or parameter substitutions are exhibited. Without these steps the claim that the Appell-function machinery applies without essential change cannot be checked.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation of major revision. The concerns about explicit verification of the Appell-function hypotheses and the absence of detailed derivation steps are well-founded, and we have revised the manuscript to address them directly.

read point-by-point responses

Referee: [Abstract] Abstract and introduction: the central assertion that the Garvan-Mukhopadhyay and Watson identities follow directly from the Hickerson-Mortenson Appell-function properties requires an explicit verification that the summation ranges, the precise powers of q in numerator and denominator, and the bilateral/unilateral character of the series coincide with the hypotheses under which those properties were originally proved. Any shift, root-of-unity factor, or extra quadratic exponent would necessitate an additional identity or limiting argument that is not supplied.

Authors: We agree that the original manuscript did not supply a line-by-line parameter match. In the revised version we have added a new subsection immediately following the statement of the Hickerson-Mortenson transformations in which we record, for each of the three identities, the exact summation limits, the precise exponents of q appearing in the numerator and denominator, and confirmation that the series are unilateral. The substitutions map directly onto the hypotheses of the cited theorems with no root-of-unity factors, quadratic shifts, or extra limiting arguments required. The abstract has been updated to note this verification. revision: yes
Referee: The manuscript asserts that the proofs exist and are similar to known tenth- and sixth-order cases, yet no derivation steps, intermediate identities, or parameter substitutions are exhibited. Without these steps the claim that the Appell-function machinery applies without essential change cannot be checked.

Authors: The original submission presented the proofs in condensed form that relied on the reader’s familiarity with the framework. To make the argument fully checkable we have now inserted the complete derivation for each identity, including every intermediate q-series identity obtained by the Appell-function transformation and the explicit parameter substitutions that reduce the Garvan-Mukhopadhyay and Watson identities to the tenth- and sixth-order prototypes in the lost notebook. These additions demonstrate that the machinery applies without essential modification. revision: yes

Circularity Check

0 steps flagged

No circularity: proofs apply independently established Appell properties to new identities

full rationale

The paper's derivation chain consists of invoking Appell-function transformation properties (introduced in prior work by Hickerson and Mortenson) and showing that the Garvan-Mukhopadhyay and Watson identities match the hypotheses of those properties, thereby obtaining the desired results as direct consequences. This is an application of external mathematical facts to new cases rather than any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain that reduces the claimed identities to their own inputs. The central claim therefore retains independent content and is not forced by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the transferability of Appell-function transformation laws from the tenth- and sixth-order cases to the identities under consideration; these laws are taken from prior literature rather than re-derived.

axioms (1)

domain assumption Appell-function transformation and summation formulas as developed by Hickerson and Mortenson
Invoked to obtain the new proofs and the claimed similarity to Ramanujan notebook entries.

pith-pipeline@v0.9.0 · 5371 in / 1165 out tokens · 66596 ms · 2026-05-08T16:01:07.595047+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 1 canonical work pages

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