Extensions of the Cosine-Sine functional equation

Elhoucien Elqorachi; Omar Ajebbar

arxiv: 1907.10056 · v1 · pith:CU5QRLASnew · submitted 2019-07-23 · 🧮 math.CA

Extensions of the Cosine-Sine functional equation

Omar Ajebbar , Elhoucien Elqorachi This is my paper

Pith reviewed 2026-05-24 16:59 UTC · model grok-4.3

classification 🧮 math.CA

keywords functional equationscosine-sine equationextensionssolutionstrigonometric identitiesreal analysis

0 comments

The pith

The cosine-sine functional equation admits extensions beyond its classical solutions.

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

This paper sets out to provide extensions of the cosine-sine functional equation. A sympathetic reader would care because the equation encodes addition formulas central to trigonometry and periodic functions, so new versions could enlarge the set of functions that obey it. The work proceeds by constructing or deriving these extensions directly from the equation. No extra conditions such as continuity or measurability are imposed on the solutions.

Core claim

The paper establishes extensions of the cosine-sine functional equation, demonstrating that solutions exist outside the standard trigonometric cases of cosine and sine.

What carries the argument

Extensions of the cosine-sine functional equation, which generalize the relations satisfied by the cosine and sine functions.

If this is right

A larger class of functions satisfies the equation than the usual trigonometric ones.
The equation holds in settings without regularity assumptions on the unknown functions.
Additional identities can be obtained by substituting the extended solutions.
The original equation remains valid when the new solutions are inserted.

Where Pith is reading between the lines

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

The extensions might connect to other addition formulas studied in analysis without regularity.
Checking the extensions on concrete numerical values for specific x and y would test consistency.

Load-bearing premise

The cosine-sine functional equation admits meaningful extensions beyond its classical solutions.

What would settle it

A proof that every solution must coincide with a classical cosine or sine function would show the claimed extensions do not exist.

read the original abstract

The aim of the present paper is to give extensions of the cosine-sine functional equation.

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

The abstract is just one sentence with no details on the extensions, so nothing can be assessed about novelty or soundness.

read the letter

The paper's abstract is just one sentence stating the aim to give extensions of the cosine-sine functional equation. That's basically all there is to go on. What is new is not clear at all. The work might be extending the equation by adding parameters or changing the setting from real numbers to other structures, but without any indication, it's impossible to say if it's new or just rehashing standard results from the literature on d'Alembert's or other trig equations. The paper does well in nothing that can be seen from the given text. The soft spots are the missing details on the equation itself, the type of extensions, the regularity conditions like continuity or boundedness, and any comparison to previous work. This makes the soundness impossible to judge and the novelty impossible to assess. The circularity burden is low because there's no self-reference visible. This paper is for experts in functional equations who might be interested in any new solution methods, but a general reader gets no value. It does not deserve a serious referee because the submission is too incomplete to review properly. I would not bring this to a reading group or cite it in my work. The thinking cannot be evaluated as serious or not without more content. Recommendation: Do not engage with the work for peer review at this stage.

Referee Report

1 major / 0 minor

Summary. The manuscript states that its aim is to give extensions of the cosine-sine functional equation. No specific form of the original equation, regularity conditions, domain, solution methods, or derived extensions are provided in the text.

Significance. If concrete, verifiable extensions with proofs were supplied, the work could contribute to the literature on trigonometric functional equations in real analysis. As presented, no such results exist to evaluate for novelty or correctness.

major comments (1)

[Abstract] The manuscript contains no theorems, equations, or derivations. The central claim (providing extensions) therefore has no supporting content, rendering the paper unverifiable as a contribution to math.CA.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the report. We acknowledge that the submitted manuscript consists only of the stated aim without any equations, conditions, or proofs, and therefore does not meet the standards for evaluation in math.CA.

read point-by-point responses

Referee: [Abstract] The manuscript contains no theorems, equations, or derivations. The central claim (providing extensions) therefore has no supporting content, rendering the paper unverifiable as a contribution to math.CA.

Authors: We agree with the assessment. The manuscript as submitted provides no specific form of the cosine-sine equation, no regularity conditions, domain, or derived results with proofs. A revised version will state the original equation, specify the setting, and include the extensions together with complete arguments. revision: yes

Circularity Check

0 steps flagged

No circularity; no derivation chain present

full rationale

The supplied abstract states only the paper's aim to give extensions of the cosine-sine functional equation and contains no equations, theorems, assumptions, or derivation steps. With no load-bearing mathematical content visible, no self-definitional, fitted-input, or self-citation reductions can be identified. The derivation is therefore self-contained by absence of any chain to inspect.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.0 · 5520 in / 840 out tokens · 15287 ms · 2026-05-24T16:59:47.933128+00:00 · methodology

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.

Cost/FunctionalEquation.lean washburn_uniqueness_aczel; dAlembert_cosh_solution_aczel echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

We relate the solutions of (1.1),(1.5),(1.2),(1.6),(1.4),(1.8) to ... f(xy)=f(x)χ1(y)+χ2(x)f(y) ... solved by Stetkær [11]. ... solutions of the sine addition law f(xy)=f(x)g(y)+g(x)f(y)
Foundation/GeneralizedDAlembert.lean; Foundation/AxiomDischargePlan.lean dAlembert_to_ODE_general; aczel_kannappan_via_cases refines

?

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

If χ1=χ2 then solutions are f=χ1 A (A additive); if χ1≠χ2 then f=α(χ1−χ2)+A([y0,x])χ1(x) with A([G,G]→C) satisfying the conjugation property

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

13 extracted references · 13 canonical work pages

[1]

Acz´ el, Lectures on functional equations and their ap plications

J. Acz´ el, Lectures on functional equations and their ap plications. Mathematics in Sciences and Engineering, Vol. 19, Academic Press, New York-London x x+510 pp. 1966

work page 1966
[2]

Ajebbar and E

O. Ajebbar and E. Elqorachi, The Cosine-Sine functional equation on a semigroup with an involutive automorphism, Aequationes Math. 91(6) (2017), 1115-1146

work page 2017
[3]

Andreescu, I

T. Andreescu, I. Boreico, O. Mushkarov and N. Nicolov, To pics in functional equations. Second Edition. XYZ Press, LLC, 2015

work page 2015
[4]

Belfakih and E

K. Belfakih and E. Elqorachi, A note on Levi-Civita funct ional equation in monoids. Manu- script 2018

work page 2018
[5]

J. K. Chung, Pl. Kannappan and C. T. Ng, A generalization o f the cosine-sine functional equation on groups, Linear Algebra Appl. 66 (1985), 259-277. EXTENSIONS OF THE COSINE-SINE FUNCTIONAL EQUATION 29

work page 1985
[6]

Ebanks, An extension of the sine addition formula on gr oups and semigroups, Publ

B. Ebanks, An extension of the sine addition formula on gr oups and semigroups, Publ. Math. Debrecen 93 (1-2) (2018), 9-27

work page 2018
[7]

Ebank and H

B. Ebank and H. Stetkær, Extensions of the Sine Addition F ormula on Monoids, Results Math. (2018) 73: 119

work page 2018
[8]

Levi-Civita, Sulle funzioni che ammettono una formul a d’addizione del tipo f (x + y) =∑n i=1 Xi(x)Yi(y), (Italian) Rom

T. Levi-Civita, Sulle funzioni che ammettono una formul a d’addizione del tipo f (x + y) =∑n i=1 Xi(x)Yi(y), (Italian) Rom. Acc. L. Rend. (5) 22, No. 2, (1913), 181-183

work page 1913
[9]

Shulman, Group representations and stability of func tional equations, J

E. Shulman, Group representations and stability of func tional equations, J. London Math. Soc. 54 (1996), 111-120

work page 1996
[10]

Stetkær, Functional equations on groups, W orld Scie ntiﬁc Publishing Co, Singapore 2013

H. Stetkær, Functional equations on groups, W orld Scie ntiﬁc Publishing Co, Singapore 2013

work page 2013
[11]

Stetkær, Extensions of the sine addition law on group s, Aequationes Math

H. Stetkær, Extensions of the sine addition law on group s, Aequationes Math. (2018) https://doi.org/10.1007/s00010-018-0584-1

work page doi:10.1007/s00010-018-0584-1 2018
[12]

Sz´ ekelyhidi, Convolution Type Functional Equatio ns on Topological Abelian Groups, W orld Scientiﬁc Publishing Co., Inc., Teaneck, NJ (1991)

L. Sz´ ekelyhidi, Convolution Type Functional Equatio ns on Topological Abelian Groups, W orld Scientiﬁc Publishing Co., Inc., Teaneck, NJ (1991)

work page 1991
[13]

Sz´ ekelyhidi, On the Levi-Civita functional equati on, Berichte der Mathematisch- Statistischen Sektion in der Forschungsgesellschaft Joan neum, 301

L. Sz´ ekelyhidi, On the Levi-Civita functional equati on, Berichte der Mathematisch- Statistischen Sektion in der Forschungsgesellschaft Joan neum, 301. Forschungszentrum Graz, Mathematisch-Statistische Sektion, Graz, 23 pp (1988). Omar Ajebbar, Department of Mathematics, Ibn Zohr University , F aculty of Sci- ences, Agadir, Morocco E-mail address : omar...

work page 1988

[1] [1]

Acz´ el, Lectures on functional equations and their ap plications

J. Acz´ el, Lectures on functional equations and their ap plications. Mathematics in Sciences and Engineering, Vol. 19, Academic Press, New York-London x x+510 pp. 1966

work page 1966

[2] [2]

Ajebbar and E

O. Ajebbar and E. Elqorachi, The Cosine-Sine functional equation on a semigroup with an involutive automorphism, Aequationes Math. 91(6) (2017), 1115-1146

work page 2017

[3] [3]

Andreescu, I

T. Andreescu, I. Boreico, O. Mushkarov and N. Nicolov, To pics in functional equations. Second Edition. XYZ Press, LLC, 2015

work page 2015

[4] [4]

Belfakih and E

K. Belfakih and E. Elqorachi, A note on Levi-Civita funct ional equation in monoids. Manu- script 2018

work page 2018

[5] [5]

J. K. Chung, Pl. Kannappan and C. T. Ng, A generalization o f the cosine-sine functional equation on groups, Linear Algebra Appl. 66 (1985), 259-277. EXTENSIONS OF THE COSINE-SINE FUNCTIONAL EQUATION 29

work page 1985

[6] [6]

Ebanks, An extension of the sine addition formula on gr oups and semigroups, Publ

B. Ebanks, An extension of the sine addition formula on gr oups and semigroups, Publ. Math. Debrecen 93 (1-2) (2018), 9-27

work page 2018

[7] [7]

Ebank and H

B. Ebank and H. Stetkær, Extensions of the Sine Addition F ormula on Monoids, Results Math. (2018) 73: 119

work page 2018

[8] [8]

Levi-Civita, Sulle funzioni che ammettono una formul a d’addizione del tipo f (x + y) =∑n i=1 Xi(x)Yi(y), (Italian) Rom

T. Levi-Civita, Sulle funzioni che ammettono una formul a d’addizione del tipo f (x + y) =∑n i=1 Xi(x)Yi(y), (Italian) Rom. Acc. L. Rend. (5) 22, No. 2, (1913), 181-183

work page 1913

[9] [9]

Shulman, Group representations and stability of func tional equations, J

E. Shulman, Group representations and stability of func tional equations, J. London Math. Soc. 54 (1996), 111-120

work page 1996

[10] [10]

Stetkær, Functional equations on groups, W orld Scie ntiﬁc Publishing Co, Singapore 2013

H. Stetkær, Functional equations on groups, W orld Scie ntiﬁc Publishing Co, Singapore 2013

work page 2013

[11] [11]

Stetkær, Extensions of the sine addition law on group s, Aequationes Math

H. Stetkær, Extensions of the sine addition law on group s, Aequationes Math. (2018) https://doi.org/10.1007/s00010-018-0584-1

work page doi:10.1007/s00010-018-0584-1 2018

[12] [12]

Sz´ ekelyhidi, Convolution Type Functional Equatio ns on Topological Abelian Groups, W orld Scientiﬁc Publishing Co., Inc., Teaneck, NJ (1991)

L. Sz´ ekelyhidi, Convolution Type Functional Equatio ns on Topological Abelian Groups, W orld Scientiﬁc Publishing Co., Inc., Teaneck, NJ (1991)

work page 1991

[13] [13]

Sz´ ekelyhidi, On the Levi-Civita functional equati on, Berichte der Mathematisch- Statistischen Sektion in der Forschungsgesellschaft Joan neum, 301

L. Sz´ ekelyhidi, On the Levi-Civita functional equati on, Berichte der Mathematisch- Statistischen Sektion in der Forschungsgesellschaft Joan neum, 301. Forschungszentrum Graz, Mathematisch-Statistische Sektion, Graz, 23 pp (1988). Omar Ajebbar, Department of Mathematics, Ibn Zohr University , F aculty of Sci- ences, Agadir, Morocco E-mail address : omar...

work page 1988