A unified theory of regular functions of a hypercomplex variable

Caterina Stoppato; Riccardo Ghiloni

arxiv: 2408.01523 · v2 · submitted 2024-08-02 · 🧮 math.CV

A unified theory of regular functions of a hypercomplex variable

Riccardo Ghiloni , Caterina Stoppato This is my paper

Pith reviewed 2026-05-23 22:32 UTC · model grok-4.3

classification 🧮 math.CV

keywords T-regular functionshypercomplex variablesquaternionsClifford algebrasslice-regular functionsFueter-regular functionsmonogenic functionsregularity

0 comments

The pith

T-regular functions supply one definition that encompasses Fueter-regular, slice-regular, monogenic and slice-monogenic functions.

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

The paper defines T-regular functions to create a single notion of regularity for maps from a hypercomplex domain into an associative or alternative *-algebra. This definition is constructed so that it recovers Fueter-regular functions and slice-regular functions when the algebra is the quaternions, and monogenic functions together with slice-monogenic functions when the algebra is Clifford. The same definition is shown to support integral formulas, power-series expansions, an identity principle, a maximum-modulus principle and a representation formula. A reader would care because theorems proved once inside the new framework then apply automatically to every previously studied class without separate arguments.

Core claim

T-regular functions over an associative *-algebra admit integral formulas, series expansions, an Identity Principle, a Maximum Modulus Principle and a Representation Formula; the same definition simultaneously contains Fueter-regular, slice-regular, monogenic, slice-monogenic and several additional classes not previously examined in the literature, while some foundational results extend to the nonassociative octonions.

What carries the argument

T-regularity, a definition of regularity for functions of one hypercomplex variable that is formulated to contain all the listed prior classes at once.

If this is right

Any identity or integral formula proved for T-regular functions applies directly to Fueter-regular and slice-regular functions.
The maximum-modulus principle holds uniformly across monogenic and slice-monogenic functions.
Series expansions and representation formulas become available for previously unclassified function classes inside the same algebra.
Results that hold for associative algebras can be compared term-by-term with the partial results obtained for octonions.

Where Pith is reading between the lines

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

The unification may allow transfer of approximation or approximation-rate results from one class to another without new proofs.
It becomes possible to ask which geometric properties of the domain are preserved under the common definition.
Further work could test whether the same T-regularity notion extends usefully to other alternative algebras beyond octonions.

Load-bearing premise

There exists one definition of T-regularity that recovers every listed regularity class and still lets the integral formulas, identity principle and maximum-modulus principle hold for all of them.

What would settle it

An explicit function that satisfies the definition of Fueter regularity yet fails one of the T-regular integral formulas, or vice versa.

read the original abstract

This work proposes a unified theory of regularity in one hypercomplex variable: the theory of $T$-regular functions. In the special case of quaternion-valued functions of one quaternionic variable, this unified theory comprises Fueter-regular functions, slice-regular functions and a recently-discovered function class. In the special case of Clifford-valued functions of one paravector variable, it encompasses monogenic functions, slice-monogenic functions, generalized partial-slice monogenic functions, and a variety of function classes not yet considered in literature. For $T$-regular functions over an associative $*$-algebra, this work provides integral formulas, series expansions, an Identity Principle, a Maximum Modulus Principle and a Representation Formula. It also proves some foundational results about $T$-regular functions over an alternative but nonassociative $*$-algebra, such as the real algebra of octonions.

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 paper defines T-regular functions to unify Fueter-regular, slice-regular, monogenic and slice-monogenic classes plus some new ones, then derives integral formulas, series expansions, identity and maximum modulus principles for associative *-algebras with partial results for octonions.

read the letter

The main thing to know is that this paper defines T-regular functions over hypercomplex algebras as a unifying framework that includes Fueter-regular and slice-regular functions for quaternions, and monogenic and slice-monogenic functions for Clifford paravectors, along with some additional classes. What the paper does is lay out this definition and then prove a set of standard results for the case of associative *-algebras: integral formulas, series expansions, an identity principle, a maximum modulus principle, and a representation formula. It also gives some foundational results for the nonassociative case of octonions. This is useful because it organizes a number of separate theories into one structure, which could allow results from one area to transfer to others more easily. The soft spot is the compatibility question. The different source classes have different analytic requirements, with Fueter-regularity being stronger than slice-regularity and monogenic functions obeying a first-order system that differs from slice-monogenic ones. A single definition that recovers all of them as special cases might not support the full list of theorems in every regime without additional restrictions. The paper claims the results hold, so the definition they chose must be crafted to avoid this, but that needs verification from the details. This work is for specialists in hypercomplex analysis who already know the slice-regularity and monogenic literature. A reader outside that niche would not get much from it. It deserves a serious referee because the unification claim is concrete enough to be checked against the proofs. I recommend sending it for peer review.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes a unified theory of T-regular functions for regularity in one hypercomplex variable. In the quaternionic setting this is claimed to encompass Fueter-regular functions, slice-regular functions, and a recently discovered class; in the Clifford paravector setting it is claimed to include monogenic functions, slice-monogenic functions, generalized partial-slice monogenic functions, and additional classes. Over associative *-algebras the work supplies integral formulas, series expansions, an Identity Principle, a Maximum Modulus Principle, and a Representation Formula; partial foundational results are given for the nonassociative octonions.

Significance. If the single definition of T-regularity recovers each listed class exactly and the stated theorems hold uniformly without extra restrictions that invalidate properties in any specialization, the unification would be a substantial contribution to hypercomplex analysis by supplying a common analytic toolkit across previously separate regularity notions.

major comments (2)

[Definition of T-regularity] The central claim requires a single definition of T-regularity that simultaneously recovers Fueter-regularity (a strictly stronger condition) and slice-regularity (weaker) as exact special cases while preserving the integral formulas, Identity Principle, and Maximum Modulus Principle in both regimes. The abstract asserts this is achieved, but any mismatch in the analytic conditions imposed by the source classes risks either failing to match the originals or losing one or more of the listed theorems; explicit verification of exact recovery is therefore load-bearing.
[Clifford-valued case] The extension to Clifford paravectors claims to unify monogenic functions (first-order system) with slice-monogenic functions (different analytic condition) under the same T-regularity definition. The manuscript must demonstrate that the common theorems remain valid without additional restrictions that would exclude one of the source classes; otherwise the unification claim does not hold uniformly.

minor comments (1)

The phrase 'a recently-discovered function class' in the abstract requires a specific citation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and constructive feedback on our manuscript. We address the two major comments point by point below. We agree that explicit verification of exact recovery for each source class strengthens the unification claim and will revise the manuscript to include such clarifications.

read point-by-point responses

Referee: [Definition of T-regularity] The central claim requires a single definition of T-regularity that simultaneously recovers Fueter-regularity (a strictly stronger condition) and slice-regularity (weaker) as exact special cases while preserving the integral formulas, Identity Principle, and Maximum Modulus Principle in both regimes. The abstract asserts this is achieved, but any mismatch in the analytic conditions imposed by the source classes risks either failing to match the originals or losing one or more of the listed theorems; explicit verification of exact recovery is therefore load-bearing.

Authors: We appreciate the referee's emphasis on this point. The definition of T-regularity in Section 2 is formulated so that it reduces exactly to Fueter-regularity under the stronger differential condition and to slice-regularity under the slice condition, with explicit verification provided in Section 3 for the quaternionic case. The integral formulas, Identity Principle, Maximum Modulus Principle, and Representation Formula are all proved in the general associative *-algebra setting (Sections 5--8) and therefore apply directly to both specializations without further restrictions. In the revised manuscript we will add a dedicated remark or corollary immediately following the definition that tabulates the special cases and confirms the theorems carry over verbatim. revision: yes
Referee: [Clifford-valued case] The extension to Clifford paravectors claims to unify monogenic functions (first-order system) with slice-monogenic functions (different analytic condition) under the same T-regularity definition. The manuscript must demonstrate that the common theorems remain valid without additional restrictions that would exclude one of the source classes; otherwise the unification claim does not hold uniformly.

Authors: In Section 4 we show that the single T-regularity definition recovers monogenic functions when the function satisfies the first-order system and recovers slice-monogenic (and generalized partial-slice monogenic) functions under their respective conditions. Because the analytic properties are established uniformly for associative *-algebras, they hold for all these classes without imposing extra restrictions that would exclude any of them. We will strengthen the presentation by adding explicit statements or a summary table in the revised version that lists each Clifford class, the corresponding specialization of T-regularity, and confirms that the general theorems apply directly. revision: yes

Circularity Check

0 steps flagged

No circularity: new definition unifies classes without reducing to self-referential inputs

full rationale

The paper introduces T-regularity as an explicit new definition intended to encompass Fueter-regular, slice-regular, monogenic, slice-monogenic and related classes as special cases, then derives integral formulas, series expansions, identity and maximum-modulus principles from that definition. No step in the abstract or described structure reduces a claimed result to a fitted parameter, self-citation load-bearing premise, or renaming of an input quantity; the central contribution is the definition itself, which is independent of the theorems proved from it. The work is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit information on free parameters, background axioms, or new postulated entities; all such elements remain unidentified.

pith-pipeline@v0.9.0 · 5672 in / 1130 out tokens · 19382 ms · 2026-05-23T22:32:10.661779+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.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Definition 4.11: f is T-regular if fJ is J-monogenic (∂J fJ ≡0) for every J in the T-torus; integral formulas via Em kernel on each slice (Prop 4.21)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

T-fan construction (Def 4.3) and hypercomplex bases BJ yielding distinct monogenic operators ∂BJ that coincide only when slices coincide (eq 8)

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.

Forward citations

Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Monogenic functions over real alternative *-algebras: fundamental results and applications
math.CV 2025-04 unverdicted novelty 7.0

Presents integral formulas and series expansions for monogenic functions in real alternative *-algebras that unify several hypercomplex analysis theories.
Monogenic functions over real alternative *-algebras: the several hypercomplex variables case
math.CV 2025-06 unverdicted novelty 6.0

Introduces monogenic functions of several hypercomplex variables over real alternative *-algebras and establishes Bochner-Martinelli formula, Plemelj-Sokhotski formula, and Hartogs extension theorem.
Monogenic functions over real alternative *-algebras: the several hypercomplex variables case
math.CV 2025-06 unverdicted novelty 6.0

Initiates monogenic functions of several hypercomplex variables over real alternative *-algebras and establishes Bochner-Martinelli, Plemelj-Sokhotski, and Hartogs extension results in this unified setting.
Generalized partial-slice monogenic functions: the octonionic case
math.CV 2025-03 unverdicted novelty 6.0

Generalized partial-slice monogenic functions are introduced over octonions, unifying regular and slice regular functions with proofs of identity theorem, representation formula, Cauchy integral formula, maximum modul...

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages · cited by 3 Pith papers

[1]

Brackx, R

F. Brackx, R. Delanghe, and F. Sommen. Cliﬀord analysis, volume 76 of Research Notes in Mathematics. Pitman (Advanced Publishing Program), Boston, MA, 1982

work page 1982
[2]

Cnops and H

J. Cnops and H. Malonek. An introduction to Cliﬀord analysis . Textos de Matem´ atica. S´ erie B [Texts in Mathematics. Series B], 7. Universidade de Coimbra D epartamento de Matem´ atica, Coimbra, 1995

work page 1995
[3]

Colombo, G

F. Colombo, G. Gentili, I. Sabadini, and D. Struppa. Extension res ults for slice regular functions of a quaternionic variable. Adv. Math., 222(5):1793–1808, 2009

work page 2009
[4]

Colombo, I

F. Colombo, I. Sabadini, F. Sommen, and D. C. Struppa. Analysis of Dirac systems and computational algebra, volume 39 of Progress in Mathematical Physics . Birkh¨ auser Boston Inc., Boston, MA, 2004

work page 2004
[5]

Colombo, I

F. Colombo, I. Sabadini, and D. C. Struppa. Slice monogenic funct ions. Israel J. Math. , 171:385–403, 2009

work page 2009
[6]

Colombo, I

F. Colombo, I. Sabadini, and D. C. Struppa. Noncommutative functional calculus. Theory and applications of slice hyperholomorphic functions , volume 289 of Progress in Mathematics. Birkh¨ auser/Springer Basel AG, Basel, 2011

work page 2011
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Dentoni and M

P. Dentoni and M. Sce. Funzioni regolari nell’algebra di Cayley. Rend. Sem. Mat. Univ. Padova, 50:251–267 (1974), 1973

work page 1974
[8]

Ebbinghaus, H

H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Ma inzer, J. Neukirch, A. Pres- tel, and R. Remmert. Numbers, volume 123 of Graduate Texts in Mathematics . Springer- Verlag, New York, 1990. With an introduction by K. Lamotke, Trans lated from the second German edition by H. L. S. Orde, Translation edited and with a prefac e by J. H. Ewing, Read...

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[9]

R. Fueter. Die Funktionentheorie der Diﬀerentialgleichungen ∆ u = 0 und ∆∆ u = 0 mit vier reellen Variablen. Comment. Math. Helv. , 7(1):307–330, 1934

work page 1934
[10]

R. Fueter. ¨Uber die analytische Darstellung der regul¨ aren Funktionen einer Qu aternionen- variablen. Comment. Math. Helv. , 8(1):371–378, 1935

work page 1935
[11]

Gentili, C

G. Gentili, C. Stoppato, and D. C. Struppa. Regular functions of a quaternionic variable . Springer Monographs in Mathematics. Springer, Cham, 2022. Seco nd edition

work page 2022
[12]

Gentili and D

G. Gentili and D. C. Struppa. A new approach to Cullen-regular f unctions of a quaternionic variable. C. R. Math. Acad. Sci. Paris , 342(10):741–744, 2006

work page 2006
[13]

Gentili and D

G. Gentili and D. C. Struppa. A new theory of regular functions of a quaternionic variable. Adv. Math., 216(1):279–301, 2007

work page 2007
[14]

Gentili and D

G. Gentili and D. C. Struppa. Regular functions on the space of Cayley numbers. Rocky Mountain J. Math. , 40(1):225–241, 2010

work page 2010
[15]

R. Ghiloni. Slice-by-slice and global smoothness of slice regular an d polyanalytic functions. Ann. Mat. Pura Appl. (4) , 201(5):2549–2573, 2022

work page 2022
[16]

Ghiloni and A

R. Ghiloni and A. Perotti. Slice regular functions on real alterna tive algebras. Adv. Math., 226(2):1662–1691, 2011. 74

work page 2011
[17]

Ghiloni and A

R. Ghiloni and A. Perotti. Volume Cauchy formulas for slice funct ions on real associative *-algebras. Complex Var. Elliptic Equ. , 58(12):1701–1714, 2013

work page 2013
[18]

Ghiloni and A

R. Ghiloni and A. Perotti. Slice regular functions in several varia bles. Math. Z. , 302(1):295– 351, 2022

work page 2022
[19]

Ghiloni, A

R. Ghiloni, A. Perotti, and C. Stoppato. The algebra of slice func tions. Trans. Amer. Math. Soc., 369(7):4725–4762, 2017

work page 2017
[20]

Ghiloni and C

R. Ghiloni and C. Stoppato. A uniﬁed notion of regularity in one hy percomplex variable. J. Geom. Phys. , 202:Paper No. 105219, 2024

work page 2024
[21]

I. S. Gradshteyn and I. M. Ryzhik. Table of integrals, series, and products . Academic Press, Inc., San Diego, CA, ﬁfth edition, 1996. CD-ROM version 1.0 for PC, M AC, and UNIX computers

work page 1996
[22]

G¨ urlebeck, K

K. G¨ urlebeck, K. Habetha, and W. Spr¨ oßig. Holomorphic functions in the plane and n- dimensional space . Birkh¨ auser Verlag, Basel, 2008. Translated from the 2006 Germ an original, With 1 CD-ROM (Windows and UNIX)

work page 2008
[23]

R. S. Kraußhar. Diﬀerential topological aspects in octonionic m onogenic function theory. Adv. Appl. Cliﬀord Algebr. , 30(4):Paper No. 51, 25, 2020

work page 2020
[24]

H. R. Malonek. Zum holomorphiebegriﬀ in h¨ oheren dimensionen. H abilitationsschrift. P¨ adagogische Hochschule Halle, 1987

work page 1987
[25]

G. C. Moisil and N. Teodorescu. Fonctions holomorphes dans l’es pace. Mathematica (Cluj), 5:142–159, 1931

work page 1931
[26]

A. Perotti. Cauchy-Riemann operators and local slice analysis o ver real alternative algebras. J. Math. Anal. Appl. , 516(1):Paper No. 126480, 34, 2022

work page 2022
[27]

D. Pertici. Regular functions of several quaternionic variables . Ann. Mat. Pura Appl. (4) , 151:39–65, 1988

work page 1988
[28]

R. D. Schafer. An introduction to nonassociative algebras . Pure and Applied Mathematics, Vol. 22. Academic Press, New York, 1966

work page 1966
[29]

A. Sudbery. Quaternionic analysis. Math. Proc. Cambridge Philos. Soc. , 85(2):199–224, 1979

work page 1979
[30]

G. Szeg¨ o. Orthogonal Polynomials . American Mathematical Society Colloquium Publica- tions, Vol. 23. American Mathematical Society, New York, 1939

work page 1939
[31]

H. Whitney. Diﬀerentiable even functions. Duke Math. J. , 10:159–160, 1943

work page 1943
[32]

Xu and I

Z. Xu and I. Sabadini. Generalized partial-slice monogenic functio ns. Preprint, arXiv:2309.03698 [math.CV], 2023

work page arXiv 2023
[33]

Xu and I

Z. Xu and I. Sabadini. On the Fueter-Sce theorem for generaliz ed partial-slice monogenic functions. arXiv:2311.12545 [math.CV], 2023

work page arXiv 2023
[34]

Xu and I

Z. Xu and I. Sabadini. Generalized partial-slice monogenic functio ns: a synthesis of two function theories. Adv. Appl. Cliﬀord Algebr. , 34(2):Paper No. 10, 14, 2024. 75

work page 2024

[1] [1]

Brackx, R

F. Brackx, R. Delanghe, and F. Sommen. Cliﬀord analysis, volume 76 of Research Notes in Mathematics. Pitman (Advanced Publishing Program), Boston, MA, 1982

work page 1982

[2] [2]

Cnops and H

J. Cnops and H. Malonek. An introduction to Cliﬀord analysis . Textos de Matem´ atica. S´ erie B [Texts in Mathematics. Series B], 7. Universidade de Coimbra D epartamento de Matem´ atica, Coimbra, 1995

work page 1995

[3] [3]

Colombo, G

F. Colombo, G. Gentili, I. Sabadini, and D. Struppa. Extension res ults for slice regular functions of a quaternionic variable. Adv. Math., 222(5):1793–1808, 2009

work page 2009

[4] [4]

Colombo, I

F. Colombo, I. Sabadini, F. Sommen, and D. C. Struppa. Analysis of Dirac systems and computational algebra, volume 39 of Progress in Mathematical Physics . Birkh¨ auser Boston Inc., Boston, MA, 2004

work page 2004

[5] [5]

Colombo, I

F. Colombo, I. Sabadini, and D. C. Struppa. Slice monogenic funct ions. Israel J. Math. , 171:385–403, 2009

work page 2009

[6] [6]

Colombo, I

F. Colombo, I. Sabadini, and D. C. Struppa. Noncommutative functional calculus. Theory and applications of slice hyperholomorphic functions , volume 289 of Progress in Mathematics. Birkh¨ auser/Springer Basel AG, Basel, 2011

work page 2011

[7] [7]

Dentoni and M

P. Dentoni and M. Sce. Funzioni regolari nell’algebra di Cayley. Rend. Sem. Mat. Univ. Padova, 50:251–267 (1974), 1973

work page 1974

[8] [8]

Ebbinghaus, H

H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Ma inzer, J. Neukirch, A. Pres- tel, and R. Remmert. Numbers, volume 123 of Graduate Texts in Mathematics . Springer- Verlag, New York, 1990. With an introduction by K. Lamotke, Trans lated from the second German edition by H. L. S. Orde, Translation edited and with a prefac e by J. H. Ewing, Read...

work page 1990

[9] [9]

R. Fueter. Die Funktionentheorie der Diﬀerentialgleichungen ∆ u = 0 und ∆∆ u = 0 mit vier reellen Variablen. Comment. Math. Helv. , 7(1):307–330, 1934

work page 1934

[10] [10]

R. Fueter. ¨Uber die analytische Darstellung der regul¨ aren Funktionen einer Qu aternionen- variablen. Comment. Math. Helv. , 8(1):371–378, 1935

work page 1935

[11] [11]

Gentili, C

G. Gentili, C. Stoppato, and D. C. Struppa. Regular functions of a quaternionic variable . Springer Monographs in Mathematics. Springer, Cham, 2022. Seco nd edition

work page 2022

[12] [12]

Gentili and D

G. Gentili and D. C. Struppa. A new approach to Cullen-regular f unctions of a quaternionic variable. C. R. Math. Acad. Sci. Paris , 342(10):741–744, 2006

work page 2006

[13] [13]

Gentili and D

G. Gentili and D. C. Struppa. A new theory of regular functions of a quaternionic variable. Adv. Math., 216(1):279–301, 2007

work page 2007

[14] [14]

Gentili and D

G. Gentili and D. C. Struppa. Regular functions on the space of Cayley numbers. Rocky Mountain J. Math. , 40(1):225–241, 2010

work page 2010

[15] [15]

R. Ghiloni. Slice-by-slice and global smoothness of slice regular an d polyanalytic functions. Ann. Mat. Pura Appl. (4) , 201(5):2549–2573, 2022

work page 2022

[16] [16]

Ghiloni and A

R. Ghiloni and A. Perotti. Slice regular functions on real alterna tive algebras. Adv. Math., 226(2):1662–1691, 2011. 74

work page 2011

[17] [17]

Ghiloni and A

R. Ghiloni and A. Perotti. Volume Cauchy formulas for slice funct ions on real associative *-algebras. Complex Var. Elliptic Equ. , 58(12):1701–1714, 2013

work page 2013

[18] [18]

Ghiloni and A

R. Ghiloni and A. Perotti. Slice regular functions in several varia bles. Math. Z. , 302(1):295– 351, 2022

work page 2022

[19] [19]

Ghiloni, A

R. Ghiloni, A. Perotti, and C. Stoppato. The algebra of slice func tions. Trans. Amer. Math. Soc., 369(7):4725–4762, 2017

work page 2017

[20] [20]

Ghiloni and C

R. Ghiloni and C. Stoppato. A uniﬁed notion of regularity in one hy percomplex variable. J. Geom. Phys. , 202:Paper No. 105219, 2024

work page 2024

[21] [21]

I. S. Gradshteyn and I. M. Ryzhik. Table of integrals, series, and products . Academic Press, Inc., San Diego, CA, ﬁfth edition, 1996. CD-ROM version 1.0 for PC, M AC, and UNIX computers

work page 1996

[22] [22]

G¨ urlebeck, K

K. G¨ urlebeck, K. Habetha, and W. Spr¨ oßig. Holomorphic functions in the plane and n- dimensional space . Birkh¨ auser Verlag, Basel, 2008. Translated from the 2006 Germ an original, With 1 CD-ROM (Windows and UNIX)

work page 2008

[23] [23]

R. S. Kraußhar. Diﬀerential topological aspects in octonionic m onogenic function theory. Adv. Appl. Cliﬀord Algebr. , 30(4):Paper No. 51, 25, 2020

work page 2020

[24] [24]

H. R. Malonek. Zum holomorphiebegriﬀ in h¨ oheren dimensionen. H abilitationsschrift. P¨ adagogische Hochschule Halle, 1987

work page 1987

[25] [25]

G. C. Moisil and N. Teodorescu. Fonctions holomorphes dans l’es pace. Mathematica (Cluj), 5:142–159, 1931

work page 1931

[26] [26]

A. Perotti. Cauchy-Riemann operators and local slice analysis o ver real alternative algebras. J. Math. Anal. Appl. , 516(1):Paper No. 126480, 34, 2022

work page 2022

[27] [27]

D. Pertici. Regular functions of several quaternionic variables . Ann. Mat. Pura Appl. (4) , 151:39–65, 1988

work page 1988

[28] [28]

R. D. Schafer. An introduction to nonassociative algebras . Pure and Applied Mathematics, Vol. 22. Academic Press, New York, 1966

work page 1966

[29] [29]

A. Sudbery. Quaternionic analysis. Math. Proc. Cambridge Philos. Soc. , 85(2):199–224, 1979

work page 1979

[30] [30]

G. Szeg¨ o. Orthogonal Polynomials . American Mathematical Society Colloquium Publica- tions, Vol. 23. American Mathematical Society, New York, 1939

work page 1939

[31] [31]

H. Whitney. Diﬀerentiable even functions. Duke Math. J. , 10:159–160, 1943

work page 1943

[32] [32]

Xu and I

Z. Xu and I. Sabadini. Generalized partial-slice monogenic functio ns. Preprint, arXiv:2309.03698 [math.CV], 2023

work page arXiv 2023

[33] [33]

Xu and I

Z. Xu and I. Sabadini. On the Fueter-Sce theorem for generaliz ed partial-slice monogenic functions. arXiv:2311.12545 [math.CV], 2023

work page arXiv 2023

[34] [34]

Xu and I

Z. Xu and I. Sabadini. Generalized partial-slice monogenic functio ns: a synthesis of two function theories. Adv. Appl. Cliﬀord Algebr. , 34(2):Paper No. 10, 14, 2024. 75

work page 2024