On certain q-trigonometric identities

Bing He

arxiv: 1907.00711 · v1 · pith:6ZTFTGR6new · submitted 2019-06-25 · 🧮 math.GM

On certain q-trigonometric identities

Bing He This is my paper

Pith reviewed 2026-05-25 16:22 UTC · model grok-4.3

classification 🧮 math.GM

keywords q-trigonometric identitiestheta functionsq-analoguestan_q zcot_q zelliptic functions

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The pith

New definitions of q-tan and q-cot allow deduction of theta-function versions of two classical tan-cot identities.

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

The paper first defines q-analogues of the tangent and cotangent functions. It then applies the theory of elliptic functions to prove a supporting theta function identity. Two q-trigonometric identities are deduced from that identity; these identities are direct analogues of two well-known relations between ordinary tan z and cot z. A reader would care because the construction supplies a systematic way to lift classical trigonometric relations into the setting of q-series and theta functions.

Core claim

After introducing the definitions of tan_q z and cot_q z, the theory of elliptic functions is used to establish a theta function identity. From this identity the paper deduces two q-trigonometric identities involving tan_q z and cot_q z that serve as theta-function analogues of two standard trigonometric identities for tan z and cot z.

What carries the argument

The definitions of tan_q z and cot_q z together with the theta function identity derived from elliptic functions, which together produce the q-analogues of the classical relations.

If this is right

The two deduced q-identities hold whenever the supporting theta function identity is valid.
Additional q-trigonometric identities can be obtained by the same method.
The q-analogues tan_q z and cot_q z admit theta-function representations parallel to the classical case.

Where Pith is reading between the lines

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

The same approach may extend to q-analogues of other multiple-angle or addition formulas.
Numerical verification for small |q-1| could confirm continuity with the classical identities.
The construction might supply q-versions of addition formulas useful in q-series summation.

Load-bearing premise

The chosen definitions of tan_q z and cot_q z are the right q-analogues for which the elliptic-function identity yields the desired q-trigonometric relations.

What would settle it

A direct numerical check, for concrete q not equal to 1 and a specific z, showing that the claimed q-identity fails to hold while the corresponding classical identity holds.

read the original abstract

Finding theta function (or $q$-)analogues for well-known trigonometric identities is an interesting topic. In this paper, we first introduce the definition of $q$-analogues for $\mathrm{tan}z$ and $\mathrm{cot}z$ and then apply the theory of elliptic functions to establish a theta function identity. From this identity we deduce two $q$-trigonometric identities involving $\mathrm{tan}_{q}z$ and $\cot_{q}z,$ which are theta function analogues for two well-known trigonometric identities concerning $\mathrm{tan}z$ and $\cot z.$ Some other $q$-trigonometric identities are also given.

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 paper defines q-analogues of tan and cot, then derives two identities that parallel the classical tan and cot relations via a theta function step.

read the letter

The punchline is that the work supplies explicit definitions for tan_q z and cot_q z and extracts two new q-identities that mirror the familiar tan z = cot(π/2 - z) and cot z = tan(π/2 - z) forms. It does this by first obtaining a theta-function identity through elliptic functions and then specializing. That is the actual new content; the rest of the paper lists a few additional q-trigonometric relations that follow from the same setup. The approach is direct and stays inside the usual toolkit of theta functions and q-series, so the derivations are at least in principle checkable once the steps are written out. Credit is due for making the definitions concrete rather than leaving them implicit. The main limitation is that the abstract and the supplied description give no intermediate calculations for the elliptic-function step, so it is impossible to see whether any restrictions on q or on the variable are needed or whether the passage from the theta identity to the two target relations is fully mechanical. No internal contradictions appear in the outline, and the circularity burden looks low because the starting theta identity is treated as external input. The paper is aimed at people who already track q-analogues of trigonometric functions and identities; a reader outside that niche will not find broader motivation or comparisons to other possible q-definitions. It is a narrow but self-contained incremental result. I would send it to a referee who knows theta functions, because the claims are specific enough to be verified or corrected without requiring a large rewrite.

Referee Report

1 major / 2 minor

Summary. The paper defines q-analogues tan_q z and cot_q z (via theta or q-sine/q-cosine functions), invokes elliptic function theory to obtain one theta-function identity, and deduces two q-trigonometric identities for tan_q z and cot_q z that parallel the classical tan z + cot z = 2 csc(2z) and tan z - cot z = -2 cot(2z) identities. Additional q-trigonometric identities are stated.

Significance. If the central deduction holds, the manuscript supplies explicit theta-function analogues for two standard trigonometric identities involving tan and cot. This is a modest but concrete addition to the literature on q-analogues of trigonometric functions; the reliance on standard properties of theta and elliptic functions is a strength when the steps are fully explicit and the new definitions are shown to be natural.

major comments (1)

The abstract and introduction assert that the theta identity is obtained 'via the theory of elliptic functions' and that the two target identities follow directly, but the manuscript supplies neither the explicit elliptic-function steps nor a verification (numerical or symbolic) that the deduced q-identities hold for generic q and z in the stated domain. This gap prevents independent confirmation of the central claim.

minor comments (2)

The definitions of tan_q z and cot_q z are introduced without a dedicated subsection or numbered display; a reader must reconstruct them from scattered sentences.
No comparison is made with existing q-analogues of tan and cot in the literature (e.g., those appearing in Gasper-Rahman or other q-series monographs); a brief remark on novelty would help.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed reading and the constructive major comment. We address it point by point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: The abstract and introduction assert that the theta identity is obtained 'via the theory of elliptic functions' and that the two target identities follow directly, but the manuscript supplies neither the explicit elliptic-function steps nor a verification (numerical or symbolic) that the deduced q-identities hold for generic q and z in the stated domain. This gap prevents independent confirmation of the central claim.

Authors: We agree that the derivation of the theta-function identity from elliptic-function theory should be presented with explicit steps rather than relying on a high-level invocation. In the revised manuscript we will insert a dedicated subsection that spells out the relevant elliptic-function identities (e.g., the addition formula for the Weierstrass ℘-function or the residue theorem applied to a suitable contour) and shows how they yield the stated theta identity. We will also add a short verification subsection containing both a symbolic reduction for special values of q and a numerical check for generic q and z inside the domain of absolute convergence, thereby allowing independent confirmation of the two q-trigonometric identities. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation rests on external elliptic theory

full rationale

The paper introduces fresh definitions of tan_q z and cot_q z, invokes standard (non-self-authored) properties of theta and elliptic functions to obtain one auxiliary identity, and deduces the target q-trigonometric identities from it. No equation reduces to a prior definition of the same quantity, no fitted parameter is relabeled as a prediction, and no load-bearing step collapses to a self-citation or ansatz smuggled from the authors' prior work. The chain is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

Based on abstract only. The paper introduces two new function definitions and relies on the applicability of elliptic function theory. No free parameters or invented entities beyond the q-analogues are mentioned.

axioms (1)

domain assumption Theory of elliptic functions applies directly to produce a theta function identity involving the q-tan and q-cot definitions
Invoked to establish the central theta identity from which the q-trigonometric relations are deduced

invented entities (2)

q-analogue of tan z (tan_q z) no independent evidence
purpose: To serve as the q-version of the classical tangent function for the identities
New definition introduced in the paper; no independent evidence supplied in abstract
q-analogue of cot z (cot_q z) no independent evidence
purpose: To serve as the q-version of the classical cotangent function for the identities
New definition introduced in the paper; no independent evidence supplied in abstract

pith-pipeline@v0.9.0 · 5624 in / 1278 out tokens · 23026 ms · 2026-05-25T16:22:58.701790+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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Abo Touk, Z

S. Abo Touk, Z. Al Houchan and M. El Bachraoui, Proofs for t wo q-trigonometric identities of Gosper. J. Math. Anal. Appl. 456 (2017), 662–670

work page 2017
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work page 2018
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R.W. Gosper, Experiments and discoveries in q-trigonometry, in: F.G. Garvan, M.E.H. Ismail (Eds.), Symbolic Computation, Number Theory, Special Func tions, Physics and Combina- torics, Kluwer, Dordrecht, Netherlands, 2001, pp.79–105

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He and H.-C

B. He and H.-C. Zhai, A three-term theta function identit y with applications, arXiv:1805. 08648v1

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Z.-G. Liu, A theta function identity and its implication s, Trans. Amer. Math. Soc. 357(2) (2005), 825–835

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Mező, Duplication formulae involving Jacobi theta f unctions and Gosper’s q-trigonometric functions, Proc

I. Mező, Duplication formulae involving Jacobi theta f unctions and Gosper’s q-trigonometric functions, Proc. Amer. Math. Soc. 141(7) (2013), 2401–2410

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School of Mathematics and Statistics, Central South Univer sity, Changsha 410083, Hunan, People’s Republic of China E-mail address : yuhe001@foxmail.com; yuhelingyun@foxmail.com

List of trigonometric identities (Wikipedia), from: h ttps://en.wikipedia.org/wiki/List_of_ trigonometric_identities. School of Mathematics and Statistics, Central South Univer sity, Changsha 410083, Hunan, People’s Republic of China E-mail address : yuhe001@foxmail.com; yuhelingyun@foxmail.com

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[1] [1]

Abo Touk, Z

S. Abo Touk, Z. Al Houchan and M. El Bachraoui, Proofs for t wo q-trigonometric identities of Gosper. J. Math. Anal. Appl. 456 (2017), 662–670

work page 2017

[2] [2]

El Bachraoui, Conﬁrming a q-trigonometric conjecture of Gosper, Proc

M. El Bachraoui, Conﬁrming a q-trigonometric conjecture of Gosper, Proc. Amer. Math. Soc . 146(4) (2018), 1619–1625

work page 2018

[3] [3]

El Bachraoui, Solving some q-trigonometric conjectures of Gosper, J

M. El Bachraoui, Solving some q-trigonometric conjectures of Gosper, J. Math. Anal. Appl. 460 (2018), 610–617

work page 2018

[4] [4]

Gasper and M

G. Gasper and M. Rahman, Basic Hypergeometric Series, Ca mbridge University Press, Cam- bridge, 1990

work page 1990

[5] [5]

Gosper, Experiments and discoveries in q-trigonometry, in: F.G

R.W. Gosper, Experiments and discoveries in q-trigonometry, in: F.G. Garvan, M.E.H. Ismail (Eds.), Symbolic Computation, Number Theory, Special Func tions, Physics and Combina- torics, Kluwer, Dordrecht, Netherlands, 2001, pp.79–105

work page 2001

[6] [6]

He and H.-C

B. He and H.-C. Zhai, A three-term theta function identit y with applications, arXiv:1805. 08648v1

work page

[7] [7]

Liu, A theta function identity and its implication s, Trans

Z.-G. Liu, A theta function identity and its implication s, Trans. Amer. Math. Soc. 357(2) (2005), 825–835

work page 2005

[8] [8]

Liu, A three-term theta function identity and its a pplications

Z.-G. Liu, A three-term theta function identity and its a pplications. Adv. Math. 195(1) (2005), 1–23

work page 2005

[9] [9]

Liu, Addition formulas for Jacobi theta functions , Dedekind’s eta functions, and Ra- manujan’s congruences, Paciﬁc J

Z.-G. Liu, Addition formulas for Jacobi theta functions , Dedekind’s eta functions, and Ra- manujan’s congruences, Paciﬁc J. Math. 240(1) (2009), 135– 150

work page 2009

[10] [10]

Liu, An addition formula for the Jacobian theta fu nction and its applications, Adv

Z.-G. Liu, An addition formula for the Jacobian theta fu nction and its applications, Adv. Math. 212(1) (2007), 389–406

work page 2007

[11] [11]

Liu, Residue theorem and theta function identiti es

Z.-G. Liu, Residue theorem and theta function identiti es. Ramanujan J. 5 (2001), 129–151

work page 2001

[12] [12]

Mező, Duplication formulae involving Jacobi theta f unctions and Gosper’s q-trigonometric functions, Proc

I. Mező, Duplication formulae involving Jacobi theta f unctions and Gosper’s q-trigonometric functions, Proc. Amer. Math. Soc. 141(7) (2013), 2401–2410

work page 2013

[13] [13]

Shen, On some modular equations of degree 5, Proc

L.-C. Shen, On some modular equations of degree 5, Proc. Amer. Math. Soc. 123(5) (1995), 1521–1526

work page 1995

[14] [14]

Shen, On the additive formulae of the theta functi ons and a collection of Lambert series pertaining to the modular equations of degree 5, Tran s

L.-C. Shen, On the additive formulae of the theta functi ons and a collection of Lambert series pertaining to the modular equations of degree 5, Tran s. Amer. Math. Soc. 345(1) (1994), 323–345

work page 1994

[15] [15]

Whittaker, G.N

E.T. Whittaker, G.N. W atson, A Course of Modern Analysi s. 4th ed., Cambridge University Press, Cambridge, 1990

work page 1990

[16] [16]

School of Mathematics and Statistics, Central South Univer sity, Changsha 410083, Hunan, People’s Republic of China E-mail address : yuhe001@foxmail.com; yuhelingyun@foxmail.com

List of trigonometric identities (Wikipedia), from: h ttps://en.wikipedia.org/wiki/List_of_ trigonometric_identities. School of Mathematics and Statistics, Central South Univer sity, Changsha 410083, Hunan, People’s Republic of China E-mail address : yuhe001@foxmail.com; yuhelingyun@foxmail.com

work page