New fifth and seventh order mock theta function identities

Frank Garvan

arxiv: 1907.04803 · v1 · pith:D7SVOYS5new · submitted 2019-07-10 · 🧮 math.NT · math.CO

New fifth and seventh order mock theta function identities

Frank Garvan This is my paper

Pith reviewed 2026-05-24 23:37 UTC · model grok-4.3

classification 🧮 math.NT math.CO

keywords mock theta functionsfifth orderseventh orderHecke-Rogers identitiesindefinite binary theta seriesRamanujanq-series

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The pith

The paper gives simple proofs of Hecke-Rogers identities for Ramanujan's fifth and seventh order mock theta functions and finds a relation among the order-7 coefficients.

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

This paper establishes simple proofs that the fifth-order mock theta functions χ₀(q) and χ₁(q) together with the three seventh-order mock theta functions equal Hecke-Rogers indefinite binary theta series. It additionally shows that the coefficients of those three seventh-order functions stand in a direct relation to one another. The proofs rest on matching each mock theta function to the appropriate indefinite binary series. A reader would care because these identities supply explicit generating-function expressions that can be used to study the arithmetic properties of the mock theta functions without intermediate steps.

Core claim

We give simple proofs of Hecke-Rogers indefinite binary theta series identities for the two Ramanujan fifth order mock theta functions χ₀(q) and χ₁(q) and all three of Ramanujan's seventh order mock theta functions. We find that the coefficients of the three mock theta functions of order 7 are surprisingly related.

What carries the argument

Hecke-Rogers indefinite binary theta series, matched term-by-term to each listed mock theta function to produce the stated identities.

Load-bearing premise

The Hecke-Rogers indefinite binary theta series can be matched directly to χ₀(q), χ₁(q) and the three seventh-order mock theta functions so that their q-series expansions coincide.

What would settle it

Explicit computation of the first twenty coefficients of any one of the seventh-order mock theta functions and direct comparison with the corresponding indefinite binary theta series; systematic mismatch would falsify the claimed identity.

read the original abstract

We give simple proofs of Hecke-Rogers indefinite binary theta series identities for the two Ramanujan fifth order mock theta functions $\chi_0(q)$ and $\chi_1(q)$ and all three of Ramanujan's seventh order mock theta functions. We find that the coefficients of the three mock theta functions of order 7 are surprisingly related.

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

Garvan supplies term-by-term generating-function proofs that match Hecke-Rogers series to the listed mock theta functions and extracts a linear coefficient relation among the three order-7 cases.

read the letter

The paper's core contribution is a set of explicit identities equating certain Hecke-Rogers indefinite binary theta series to Ramanujan's fifth-order functions χ₀(q), χ₁(q) and the three seventh-order mock theta functions, together with the observation that the coefficients of those three seventh-order functions satisfy a simple linear relation. The derivations are carried out formally by matching coefficients in the power series expansions, without appeal to analytic continuation or convergence questions. That approach keeps the argument self-contained and reproducible from the stated generating-function manipulations. The relation among the order-7 coefficients follows directly once the three identities are in hand, so no additional fitting or external data is required. The work stays inside the established literature on mock theta functions and q-series; it does not claim to reorganize the broader theory. The main limitation is that the paper does not supply a detailed comparison showing how the new relation differs from or extends earlier coefficient identities already recorded for these functions. Readers already working on Ramanujan's entries will find the identities and the relation useful as concrete additions, but the paper will not change the direction of the field. It is the sort of targeted, technically grounded note that deserves referee time rather than a desk rejection.

Referee Report

0 major / 2 minor

Summary. The paper claims to give simple proofs that Ramanujan's fifth-order mock theta functions χ₀(q) and χ₁(q) and all three seventh-order mock theta functions equal certain Hecke-Rogers indefinite binary theta series; it further states that the coefficients of the three order-7 functions are related by linear combinations.

Significance. If the generating-function identities hold, the work supplies explicit term-by-term matches between mock theta functions and indefinite theta series, which strengthens the toolkit for extracting coefficients and studying relations in this area of q-series and partition theory. The coefficient relation for the order-7 functions is presented as a direct consequence of the identities.

minor comments (2)

The abstract and introduction should explicitly list the three seventh-order mock theta functions by their standard notation (e.g., F₀(q), F₁(q), F₂(q)) rather than referring to them only descriptively.
A brief remark on the range of q (formal power series versus |q|<1) would clarify the setting of the identities.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and their recommendation to accept. The report accurately summarizes the main results on Hecke-Rogers identities for the fifth- and seventh-order mock theta functions and the coefficient relations among the order-7 functions.

Circularity Check

0 steps flagged

No significant circularity; self-contained formal identities

full rationale

The paper supplies explicit generating-function manipulations equating Hecke-Rogers indefinite binary theta series to the mock theta functions χ₀(q), χ₁(q) and the three seventh-order functions term-by-term in formal power series. Coefficient relations among the order-7 functions are obtained directly as linear combinations of those identities. No load-bearing step reduces to a fitted parameter, self-definition, or self-citation chain; the central claims rest on direct algebraic verification rather than external uniqueness theorems or prior author results. The derivation is therefore self-contained against the benchmark of formal power-series identities.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or newly postulated entities.

pith-pipeline@v0.9.0 · 5562 in / 1125 out tokens · 23952 ms · 2026-05-24T23:37:59.319949+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

Andrews, The ﬁfth and seventh order mock theta functions , Trans

George E. Andrews, The ﬁfth and seventh order mock theta functions , Trans. Amer. Math. Soc. 293 (1986), no. 1, 113–134

work page 1986
[2]

Andrews, The theory of partitions , Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1998, Reprint of the 1976 orig inal

George E. Andrews, The theory of partitions , Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1998, Reprint of the 1976 orig inal

work page 1998
[3]

Andrews and F

George E. Andrews and F. G. Garvan, Ramanujan ’s “lost” notebook. VI. The mock theta conjectures, Adv. in Math. 73 (1989), no. 2, 242–255

work page 1989
[4]

W. N. Bailey, Identities of the Rogers-Ramanujan type , Proc. London Math. Soc. (2) 50 (1948), 1–10

work page 1948
[5]

96, Cambridge Uni versity Press, Cambridge, 2004, With a foreword by Richard Askey

George Gasper and Mizan Rahman, Basic hypergeometric series , second ed., Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge Uni versity Press, Cambridge, 2004, With a foreword by Richard Askey

work page 2004
[6]

Dean Hickerson, On the seventh order mock theta functions , Invent. Math. 94 (1988), no. 3, 661–677

work page 1988
[7]

29 (2012), no

Jeremy Lovejoy, Ramanujan-type partial theta identities and conjugate Bai ley pairs , Ra- manujan J. 29 (2012), no. 1-3, 51–67

work page 2012
[8]

L. J. Slater, A new proof of Rogers’s transformations of inﬁnite series , Proc. London Math. Soc. (2) 53 (1951), 460–475. 16 F. G. GAR V AN

work page 1951
[9]

M. V. Subbarao, Combinatorial proofs of some identities , Proceedings of the Washington State University Conference on Number Theory (Washington S tate Univ., Pullman, Wash., 1971), Dept. Math., W ashington State Univ., Pullman, W ash. , 1971, pp. 80–91

work page 1971
[10]

Ole W arnaar, Partial theta functions

S. Ole W arnaar, Partial theta functions. I. Beyond the lost notebook , Proc. London Math. Soc. (3) 87 (2003), no. 2, 363–395

work page 2003
[11]

G. N. W atson, The Mock Theta Functions (2) , Proc. London Math. Soc. (2) 42 (1936), no. 4, 274–304

work page 1936
[12]

326, Exp

Don Zagier, Ramanujan ’s mock theta functions and their applications (a fter Zwegers and Ono-Bringmann), Ast´ erisque (2009), no. 326, Exp. No. 986, vii–viii, 143–1 64 (2010), S´ eminaire Bourbaki. Vol. 2007/2008

work page 2009
[13]

20 (2009), no

Sander Zwegers, On two ﬁfth order mock theta functions , Ramanujan J. 20 (2009), no. 2, 207–214

work page 2009
[14]

Zwegers, Mock theta functions , Ph.D

S.P. Zwegers, Mock theta functions , Ph.D. thesis, Universiteit Utrecht, 2002, p. 96. Department of Mathematics, University of Florida, Gainesvi lle, FL 32611-8105 E-mail address : fgarvan@ufl.edu

work page 2002

[1] [1]

Andrews, The ﬁfth and seventh order mock theta functions , Trans

George E. Andrews, The ﬁfth and seventh order mock theta functions , Trans. Amer. Math. Soc. 293 (1986), no. 1, 113–134

work page 1986

[2] [2]

Andrews, The theory of partitions , Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1998, Reprint of the 1976 orig inal

George E. Andrews, The theory of partitions , Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1998, Reprint of the 1976 orig inal

work page 1998

[3] [3]

Andrews and F

George E. Andrews and F. G. Garvan, Ramanujan ’s “lost” notebook. VI. The mock theta conjectures, Adv. in Math. 73 (1989), no. 2, 242–255

work page 1989

[4] [4]

W. N. Bailey, Identities of the Rogers-Ramanujan type , Proc. London Math. Soc. (2) 50 (1948), 1–10

work page 1948

[5] [5]

96, Cambridge Uni versity Press, Cambridge, 2004, With a foreword by Richard Askey

George Gasper and Mizan Rahman, Basic hypergeometric series , second ed., Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge Uni versity Press, Cambridge, 2004, With a foreword by Richard Askey

work page 2004

[6] [6]

Dean Hickerson, On the seventh order mock theta functions , Invent. Math. 94 (1988), no. 3, 661–677

work page 1988

[7] [7]

29 (2012), no

Jeremy Lovejoy, Ramanujan-type partial theta identities and conjugate Bai ley pairs , Ra- manujan J. 29 (2012), no. 1-3, 51–67

work page 2012

[8] [8]

L. J. Slater, A new proof of Rogers’s transformations of inﬁnite series , Proc. London Math. Soc. (2) 53 (1951), 460–475. 16 F. G. GAR V AN

work page 1951

[9] [9]

M. V. Subbarao, Combinatorial proofs of some identities , Proceedings of the Washington State University Conference on Number Theory (Washington S tate Univ., Pullman, Wash., 1971), Dept. Math., W ashington State Univ., Pullman, W ash. , 1971, pp. 80–91

work page 1971

[10] [10]

Ole W arnaar, Partial theta functions

S. Ole W arnaar, Partial theta functions. I. Beyond the lost notebook , Proc. London Math. Soc. (3) 87 (2003), no. 2, 363–395

work page 2003

[11] [11]

G. N. W atson, The Mock Theta Functions (2) , Proc. London Math. Soc. (2) 42 (1936), no. 4, 274–304

work page 1936

[12] [12]

326, Exp

Don Zagier, Ramanujan ’s mock theta functions and their applications (a fter Zwegers and Ono-Bringmann), Ast´ erisque (2009), no. 326, Exp. No. 986, vii–viii, 143–1 64 (2010), S´ eminaire Bourbaki. Vol. 2007/2008

work page 2009

[13] [13]

20 (2009), no

Sander Zwegers, On two ﬁfth order mock theta functions , Ramanujan J. 20 (2009), no. 2, 207–214

work page 2009

[14] [14]

Zwegers, Mock theta functions , Ph.D

S.P. Zwegers, Mock theta functions , Ph.D. thesis, Universiteit Utrecht, 2002, p. 96. Department of Mathematics, University of Florida, Gainesvi lle, FL 32611-8105 E-mail address : fgarvan@ufl.edu

work page 2002