Articulo 10.dvi


CUBO A Mathematical Journal
Vol.12, No¯ 02, (145–167). June 2010

On homogeneous polynomial solutions of generalized
Moisil-Théodoresco systems in Euclidean space

Richard Delanghe

Department of Mathematical Analysis,

Clifford Research Group,

Ghent University,

Galglaan 2, B-9000 Ghent, Belgium

ABSTRACT

Let for s ∈ {0, 1, ..., m + 1} (m ≥ 2) , IR
(s)
0,m+1 be the space of s-vectors in the Clifford al-

gebra IR0,m+1 constructed over the quadratic vector space IR
0,m+1 and let r, p, q, ∈ IN be

such that 0 ≤ r ≤ m + 1, p < q and r + 2q ≤ m + 1. The associated linear system of

first order partial differential equations derived from the equation ∂xW = 0 where W is

IR
(r,p,q)
0,m+1 =

∑q
j=p ⊕IR

(r+2j)
0,m+1 -valued and ∂x is the Dirac operator in IR

m+1, is called a gen-

eralized Moisil-Théodoresco system of type (r, p, q) in IRm+1. For k ∈ N, k ≥ 1, M T +(m +

1; k; IR
(r,p,q)
0,m+1), denotes the space of IR

(r,p,q)
0,m+1-valued homogeneous polynomials Wk of degree

k in IRm+1 satisfying ∂xWk = 0. A characterization of Wk ∈ M T
+(m + 1; k; IR

(r,p,q)
0,m+1) is

given in terms of a harmonic potential Hk+1 belonging to a subclass of IR
(r,p,q)
0,m -valued

solid harmonics of degree (k + 1) in IRm+1. Furthermore, it is proved that each Wk ∈

M T +(m + 1; k; IR
(r,p,q)
0,m+1) admits a primitive Wk+1 ∈ M T

+(m + 1; k + 1; IR
(r,p,q)
0,m+1). Special

attention is paid to the lower dimensional cases IR3 and IR4. In particular, a method is

developed for constructing bases for the spaces M T +(4; k; IR
(r,p,q)
0,4 ), r being even.

RESUMEN

Para s ∈ {0, 1, ..., m + 1} (m ≥ 2) , IR
(s)
0,m+1 el espacio de los s-vectors en el algebra de Clif-

ford IR0,m+1 construida sobre el espacio de vectores cuadráticos IR
0,m+1 sea r, p, q, ∈ IN

tal que 0 ≤ r ≤ m + 1, p < q. El sistema lineal asociado de ecuaciones diferenciales



146 Richard Delanghe CUBO
12, 2 (2010)

parciales de primer orden derivado de la ecuación ∂xW = 0 donde W es IR
(r,p,q)
0,m+1 =∑q

j=p ⊕IR
(r+2j)
0,m+1 -valuada y ∂x es el operador de Dirac en IR

m+1, es llamado un sistema de

Moisil-Théodoresco generalizado de tipo (r, p, q) en IRm+1. Para k ∈ N, k ≥ 1, M T +(m +

1; k; IR
(r,p,q)
0,m+1), denota el espacio de polinomios homogéneos Wk IR

(r,p,q)
0,m+1- valuados de grado

k en IRm+1 satisfaciendo ∂xWk = 0. Una caracterización de Wk ∈ M T
+(m+1; k; IR

(r,p,q)
0,m+1)

es dada en términos de un potencial armónico Hk+1 perteneciendo a una subclase de

armónicos consistentes IR
(r,p,q)
0,m -valuados de grado (k + 1) in IR

m+1. Además es probado

que todo Wk ∈ M T
+(m + 1; k; IR

(r,p,q)
0,m+1) admite una primitiva Wk+1 ∈ M T

+(m + 1; k +

1; IR
(r,p,q)
0,m+1). Una especial atención es dada a los casos de dimensión baja IR

3 y IR4. En par-

ticular, un metodo es desarrollado para construir bases para espacios M T +(4; k; IR
(r,p,q)
0,4 ), r

siendo par.

Key words and phrases: Clifford analysis; Moisil-Théodoresco systems; conjugate harmonic fun-

tions; harmonic potentials; polynomial bases

Mathematics Subject Classification (2000): 30G35

1 Introduction

Let f̃ = (f0,
→

f ) with
→

f = (f1, f2, f3) be a IR
4-valued C1-function in some oppropriate open domain Ω

of IR3.

In [20] it was pointed out that the Riesz system

(R)





div
→
f = 0

curl
→

f = 0
(1.1)

and the Moisil-Théodoresco system

(M T )





div
→

f = 0

grad f0 + curl
→
f = 0

(1.2)

are examples of natural generalizations of the Cauchy-Riemann equations in the plane to Euclid-

ean space IR3.

Obviously (1.1) may be derived from (1.2) by taking f0 = 0.

For the original definition of the (M T )-system (1.2) we refer to [17].

The (M T )-system (1.2) may also be obtained by making use of the algebra H of real quaternions.

Indeed, consider in IR3 the operator D3 = i∂x0 + j∂x1 + k∂x2 where (i, j, k) is the standard set of

imaginary units in H, and associate with (f0,
→
f ) the H-valued function f = f0 + if1 + jf2 + kf3. Then

f̃ = (f0,
→
f ) satisfying (1.2) is equivalent to f satisfying the equation (see e.g. [15])

D3f = 0 (1.3)



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On homogeneous polynomial solutions of generalized
Moisil-Théodoresco systems in Euclidean space 147

In [14] solutions to (1.3) were called H-regular.

In [20] the authors studied properties of solutions to more general first order linear systems in Euclidean

space generalizing the Cauchy-Riemann equations and having the property that they are invariant

under rotations. Among such systems figures in particular the Hodge-de Rham system

(HdR)

{
dωr = 0

d∗ωr = 0
(1.4)

ωr being a smooth r-form.

Putting IRm+1 = {x = (x0, x) : x = (x1, ..., xm) ∈ IR
m}, let us recall that for a smooth s-form ωs =∑

|A|=s ω
s
Adx

A(0 ≤ s ≤ m + 1) where for A = {i1, i2, ..., is} with 0 ≤ i1 < i2 < ... < is ≤ m, dx
A =

dxi1 ∧ dxi2 ∧ ... ∧ dxis , dωs and d∗ωs are defined by (see [16])

dωs =
∑

A

m∑

i=0

∂xi ω
s
Adx

i ∧ dxA

and (1.5)

d
∗
ω

s =
∑

A

s∑

j=1

(−1)
j∂xij ω

s
Adx

A\{ij }

Notice that if in (1.4), r = 1 and ω1 = u0dx
0 + u1dx

0 + u1dx
1 + ... + umdx

m, then





dω1 = 0

d∗ω1 = 0

⇐⇒





∂uj
∂xi

− ∂ui
∂xj

= 0, i, j = 0, ..., m; i 6= j

m∑

i=0

∂ui

∂xi
= 0

(1.6)

Obviously, (1.6) generalizes to IRm+1 the system (R) in IR3 defined by (1.1).

An (m+1)-system u = (u0, u1, ..., um) satisfying (1.6) was called in [19] a system of conjugate harmonic

funtions.

More generally, introducing the differential operator D = d + d∗, its action on the smooth differential

form ω =
∑m+1

s=0 ω
s in Ω ⊂ IRm+1 open, leads to the first order system of differential equations

(d + d∗)ω = 0 ⇐⇒





d∗ω1 = 0

dωs + d∗ωs+2 = 0, s = 0, ..., m − 1

dωm = 0

(1.7)

Obviously, if ω = ωr, 0 ≤ r ≤ m + 1 being fixed, the system (1.7) reduces to the Hodge-de Rham

system (1.4).

In IR3, associating with f̃ = (f0,
→
f ), the form ω = ω0 + ω2 with ω0 = f0 and ω

2 = f1dx
1 ∧ dx2 +



148 Richard Delanghe CUBO
12, 2 (2010)

f2dx
2 ∧ dx0 + f3dx

0 ∧ dx1, then clearly (d + d∗)ω = 0 is equivalent with (f0,
→
f ) satisfying (1.2).

Let us now describe how the system (1.7) may also be obtained within the framework of Clifford

analysis.

Let IR0,m+1 be the real vector space IRm+1 equipped with a quadratic form of signature (0, m+ 1) and

let e = (e0, e1, ..., em) be an orthogonal basis of IR
0,m+1. Furthermore, let IR0,m+1 be the universal

Clifford algebra constructed over IR0,m+1. Then IR0,m+1 is a linear associative but non-commutative

algebra with identity and having dimension 2m+1.

The basic multiplication rules in IR0,m+1 are governed by

e2i = −1, i = 0, 1, ..., m

eiej + ej ei = 0, i 6= j, i, j = 0, 1, ...m.

A basis for IR0,m+1 is given by the set (eA : A ⊂ {0, 1, ..., m} , |A| = s, s = 0, 1, ..., m + 1) where

A = {i1, ..., is} with 0 ≤ i1 < i2 < ... < is ≤ m and eA = ei1 ei2 ...eis , e∅ = 1 being the identity element

of IR0,m+1.

For s ∈ {0, 1, ..., m + 1} fixed the space IR
(s)
0,m+1 of s-vectors in IR0,m+1 is defined by IR

(s)
0,m+1 =

spanIR(eA : |A| = s).

Obviously

IR0,m+1 =

m+1∑

s=0

⊕IR
(s)
0,m+1 (1.8)

Denote by [ ]
s

the projection of IR0,m+1 onto IR
(s)
0,m+1.

For a 1-vector v and a s-vector ws, we have that vws splits into

vws = [vws]
s−1 + [vw

s]
s+1

where

[vws]
s−1 =

1

2
(vws − (−1)

s
wsv)

and (1.9)

[vws]
s+1 =

1

2
(vws + (−1)

s
wsv)

Introducing the Dirac operator ∂x =
∑m

i=0 ei∂xi in IR
m+1, then by (1.9) its (left) action on a

IR
(s)
0,m−1-valued smooth funtion W

s in Ω ⊂ IRm+1 open reads:

∂xW
s =

(
∂

+
x + ∂

−
x

)
W

s

where

∂+x W
s = [∂xW

s]
s+1 =

1

2
(∂xW

s + (−1)
s
W s∂x)



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On homogeneous polynomial solutions of generalized
Moisil-Théodoresco systems in Euclidean space 149

and (1.10)

∂
−
x W

s = [∂xW
s]

s−1 =
1

2
(∂xW

s − (−1)
s
W

s
∂x) ,

W s∂x meaning that ∂x acts from the right on W
s.

It thus follows that if the IR0,m+1 valued smooth function W in Ω is decomposed along (1.8), i.e.

W =
∑m+1

s=0 W
s with W s smooth and IR

(s)
0,m+1-valued in Ω, then

∂xW = 0 ⇐⇒





∂−x W
1 = 0

∂+x W
s + ∂−x W

s+2 = 0, s = 0, 1, ..., m − 1

∂+x W
m = 0

(1.11)

Obviously, the systems (1.7) and (1.11) show some parallelism, which becomes fully clear if one

considers the isomorphism Θ between the spaces E
(
Ω; ∧sIRm+1

)
of smooth s-forms in Ω and the space

E
(
Ω; IR

(s)
0,m+1

)
of smooth IR

(s)
0,m+1-valued functions in Ω.

Indeed, associate with ωs =
∑

|A|=s ω
s
Adx

A ∈ E
(
Ω; ∧sIRm+1

)
, Θωs = W s =

∑
|A|=s W

s
AeA ∈

E
(
Ω; IR

(s)
0,m+1

)
where for all A, W sA = ω

s
A.

Then clearly, through Θ, the action of ∂+x and ∂
−
x on E

(
Ω; IR

(s)
o,m+1

)
as given in (1.10) corresponds,

respectively, to the action of d and d∗ on E
(
Ω; ∧sIRm+1

)
.

Notice in particular that on E
(
Ω; IR

(s)
o,m+1

)
, ∂+2x = 0 and ∂

−2
x = 0 whence, as ∂

2
x = −∆x, ∆x be-

ing the Laplacian in IRm+1, we have that on E
(
Ω; IR

(s)
o,m+1

)
, ∂+x ∂

−
x + ∂

−
x ∂

+
x = −∆x.

In [5] smooth differential forms ω satisfying (d − d∗) ω = 0 in Ω were called self-conjugate. It may be

easily verified that if W = Θω, then

(d − d∗) ω = 0 ⇐⇒ W ∂x = 0. (1.12)

Now let r ∈ {0, 1, ..., m + 1} be fixed and let p, q ∈ N be such that p < q and r + 2q ≤ m + 1.

Then by IR
(r,p,q)
0,m+1 we denote the subspace of IR0,m+1 defined by

IR
(r,p,q)
0,m+1 =

q∑

j=p

⊕IR
(r+2j)
0,m+1.

For a IR
(r,p,q)
0,m+1-valued smooth function W in Ω which decomposes as W =

∑q
j=p W

r+2j, we have

(see also (1.11))

∂xW = 0 ⇐⇒





∂−x W
r+2p = 0

∂+x W
r+2j + ∂−x W

r+2(j+1) = 0, j = p, ..., q − 1

∂+x W
r+2q = 0

(1.13)

The system (1.13) is called a generalized Moisil-Théodoresco system of type (r, p, q) ((GMT)-

system of type (r, p, q)). It was introduced in [1] where some general properties of solutions to this



150 Richard Delanghe CUBO
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system have been investigated.

Of course, one could as well have considered the so-called (GMT)-system of type (r, p, q) adjoint to

(1.13) which corresponds to the equation W ∂x = 0. By virtue of (1.12), putting ω = Θ
−1W would

thus imply that ω satisfies the equation (d − d∗) ω = 0.

In the case IR3 we refer to [9] for the notions of the (MT)-system and its adjoint.

The aim of the underlying paper is to study the space M T +
(
m + 1; k; IR

(r,p,q)
0,m+1

)
of IR

(r,p,q)
0,m+1-valued

homogeneous polynomial solutions Wk of degree k (k ≥ 1) in IR
m+1 to the system (1.13).

The study of the space M T +
(
m + 1; k; IR

(r,p,q)
0,m+1

)
is motivated by the fact that if W is a solution to

(1.13) in Ω, then it is real-analytic in Ω. This implies that if e.g. O ∈ Ω, then in some open ball
◦
B (O, R) ⊂ Ω, W admits a Taylor expansion

W (x) =

∞∑

k=0

∑

|γ|=k

1

γ!
xγ ∂γ W (0),

where as usual, for γ = (γ0, γ1, ..., γm) ∈ IN
m+1

, |γ| =
∑m

i=0 γi, γ! = γ0!γ1!...γm!, x
γ = x

γ0
0 x

γ1
1 ...x

γm
m

and ∂γ = ∂γ0x0 ∂
γ1
x1

...∂γmxm .

Consequently, for each k ∈ IN, k ≥ 1,

Wk(x) =
∑

|γ|=k

1

γ!
xγ ∂γ W (0)

is an element of M T +(m + 1; k; IR
(r,p,q)
0,m+1).

The following results are obtained:

(i) Let IR0,m be the Clifford algebra constructed over IR
0,m, the latter being the subspace of IR0,m+1

spanned by e = (e1, e2, ..., em) and let IR
+
0,m+1 and IR

+
0,m be the even subalgebras of IR0,m+1 and

IR0,m (see § 2).

Then the operator Dx : H(m + 1; k + 1; IR
+
0,m) → M T

+(m + 1; k; IR+0,m+1) is surjective (Corollary

4.3). The operator Dx is the conjugate of the Cauchy-Riemann operator Dx = e0∂x in IR
m+1 where

e0 = −e0 and H(m + 1; k + 1; R
+
0,m) is the space of IR

+
0,m-valued solid harmonics of degree k + 1 in

IR
m+1.

Notice that in the case m = 1, IR+0,2 = IR ⊕ e0e1IR
∼= C (identify e0e1 with the imaginary unit),

IR
+
0,1

∼= IR and that 12 Dx =
1
2
(∂x0 − e0e1∂x1 ) is nothing else but the operator ∂z in IR

2.

Corollary 4.3 thus generalizes to IRm+1 the classical result in complex analysis stating that each homo-

geneous holomorphic polynomial of degree k is the derivative (w.r.t. ∂z ) of a real-valued homogeneous

harmonic polynomial of degree k + 1.

Corollary 4.3 in fact follows from a more general result (Theorem 4.2), the proof of which relies heavily

on the existence of conjugate harmonic pairs (Uk, Vk) as treated in section 4.1 and on a refined version

of the Poincaré Lemmas for the case of r-forms with homogeneous polynomial coefficients (see section

3).



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Moisil-Théodoresco systems in Euclidean space 151

(ii) In the case IR4, bases are constructed for the spaces M T +(4; k; IR
(r,p,q)
0,4 ), r being even (see section

5). The fact that in studying (GMT)-systems of type (r, p, q), we may restrict ourselves to the case r

even has been shown in section 2.

(iii) It is proved in section 4.3 that each Wk ∈ M T
+(m + 1; k; IR

(r,p,q)
0,m+1) admits a primitive Wk+1 ∈

M T +(m + 1; k + 1; IR
(r,p,q)
0,m+1), i.e. Wk = DxWk+1.

2 (GMT)-systems revisited

Let again IR0,m+1 be the Clifford algebra constructed over the quadratic space IR
0,m+1 with orthogonal

basis e = (e0, e1, ..., em) and let (eA : |A| = s, s = 0, 1, ..., m + 1) be the standard basis of IR0,m+1.

In (1.8), a decomposition of IR0,m+1 in terms of its r-vector subspaces IR
(r)
0,m+1, r = 0, 1, ..., m + 1, was

obtained:

IR0,m+1 =

m+1∑

r=0

⊕IR
(r)
0,m+1 (2.1)

The even subalgebra IR+0,m+1 and the odd subspace IR
−
0,m+1 of IR0,m+1 are defined by

IR
+
0,m+1 =

∑

reven

⊕IR
(r)
0,m+1

and

IR
−
0,m+1 =

∑

rodd

⊕IR
(r)
0,m+1.

Obviously

IR0,m+1 = IR
+
0,m+1 ⊕ IR

−
0,m+1 (2.2)

The conjugation a → a on IR0,m+1 is defined by the basic properties

ei = −ei, i = 0, 1, ..., m

and

ab = b a, a, b ∈ IR0,m+1

The factorization IRm+1 = IR × IRm leads to the following third decomposition of IR0,m+1.

Restrict the quadratic form on IRm+1 to IRm, thus obtaining the quadratic vector space IR0,m with

orthogonal basis e = (e1, ..., em).

Then inside IR0,m+1, IR
0,m generates the Clifford algebra IR0,m and clearly



152 Richard Delanghe CUBO
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IR0,m+1 = IR0,m ⊕ e0IR0,m (2.3)

It thus follows that

IR
+
0,m+1 = IR

+
0,m ⊕ e0IR

−
0,m

and (2.4)

IR
−
0,m+1 = IR

−
0,m ⊕ e0IR

+
0,m

Put M = {0, 1, ..., m} and
◦

M = {1, ..., m} and consider the so-called pseudo-scalars eM = e0e1...em

and e ◦
M

= e1e2...em in, respectively, IR0,m+1 and IR0,m.

If m + 1 is odd, then clearly right multiplication by eM determines an isomorphism a
− → a−eM , a

− ∈

IR
−
0,m+1, between IR

−
0,m+1 and IR

+
0,m+1. In the case m + 1 even, right multiplication by e ◦

M
determines

an isomorphism a− → a−e ◦
M

between IR−0,m+1 and IR
+
0,m+1. Indeed, by virtue of the decomposition

(2.4), we have that, if a− ∈ IR−0,m+1 is written as

a− = b− + e0b
+, b− ∈ IR−0,m, b

+ ∈ IR+0,m,

then a−e ◦
M

∈ IR+0,m+1 with a
−e ◦

M
= b−e ◦

M
+ e0b

+e ◦
M

where b−e ◦
M

∈ IR+0,m and b
+e ◦

M
∈ IR−0,m.

Now let Ω ⊂ IRm+1 be open and let W : Ω → IR0,m+1 be a C1-function in Ω. Then W is said to be

left monogenic in Ω if ∂xW = 0 in Ω where ∂x =
∑m

i=0 ei∂xi is the Dirac operator in IR
m+1.

By virtue of (2.2), W splits into

W = W + + W −

where W + and W − are R+0,m+1- and IR
−
0,m+1-valued C1-functions in Ω

Consequently

∂xW = 0 ⇐⇒

{
∂xW

+ = 0

∂xW
− = 0

(2.5)

If ∂xW = 0, then the set of components of W
+ or of W − was called in [18] a system of conjugate

harmonic functions in Ω.

Taking into account the isomorphisms IR−0,m+1 → IR
+
0,m+1 where a

− → a−eM if m + 1 is odd and

a− → a−e ◦
M

if m + 1 is even, a− ∈ IR−0,m+1, it follows from (2.5) that it suffices in fact to study left

monogenic IR+0,m+1-valued functions in Ω.

Now let W =
∑q

j=p W
r+2j be a IR

(r,p,q)
0,m+1-valued left monogenic function defined in Ω, i.e. W

satisfies the so-called generalized Moisil-Théodoresco system of type (r, p, q) (see also (1.13)).

∂xW = 0 ⇐⇒





∂−W r+2p = 0

∂+x W
r+2j + ∂−x W

r+2(jH) = 0, j = p, ..., q − 1

∂+W r+2q = 0

(2.6)



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On homogeneous polynomial solutions of generalized
Moisil-Théodoresco systems in Euclidean space 153

then in view of the observations made, it suffices to consider functions W =
∑q

j=p W
r+2j where

r is even.

However, let us point out that if W s is IR
(s)
0,m+1-valued with s odd then

(i) for m + 1 odd, W seM is IR
(m+1−s)
0,m+1 -valued

while

(ii) for m + 1 even, W se ◦
M

is IR
(m−s)
0,m+1 ⊕ IR

(m−s+2)
0,m+1 -valued

Property (i) being obvious, let us prove (ii).

Write out W s as W s =
∑

|A|=s W
s
AeA and put for A ⊂ {0, 1, ..., m} with |A| = s, A = {i1, i2, ..., is}.

If i1 = 0, then A = e0eB where B = {i2, ..., is} ⊂ {1, ..., m} and so eAe ◦
M

= e0eBe ◦
M

= ±e0e ◦
M\B

∈

IR
(m−s+2)
0,m+1 .

If i1 6= 0, then {i1, ..., is} ⊂ {1, ..., m} and eAe ◦
M

= ±e ◦
M\A

∈ IR
(m−s)
0,m ⊂ IR

(m−s)
0,m+1.

We may thus conclude that for s odd and W IR
(s,p,q)
0,m+1-valued in Ω, in the case (m + 1) odd, W eM is

IR
(s∗,0,q∗)
0,m+1 -valued where s

∗ = m + 1 − (s + 2q) and q∗ = q − p, while in the case (m + 1) even, W e ◦
M

is IR
(s∗,0,q∗)
0,m+1 -valued where s

∗ = m − s − 2q and q∗ = q − p + 1.

These properties are nicely illustrated in the cases IR3 and IR4.

In IR3, the following (r, p, q)-subspaces occur:

(i) for r even: IR
(0)
0,3 ⊕ IR

(2)
0,3 = IR

+
0,3

(ii) for r odd: IR
(1)
0,3 ⊕ IR

(3)
0,3 = IR

−
0,3

and

IR
−
0,3eM = IR

−
0,3e0e1e2 = IR

+
0,3.

In IR4, we have:

(i) for r even:

IR
(0,0,1)
0,4 = IR

(0)
0,4 ⊕ IR

(2)
0,4; IR

(2,0,1)
0,4 = IR

(2)
0,4 ⊕ IR

(4)
0,4

IR
(0,0,2)
0,4 = IR

(0)
0,4 ⊕ IR

(2)
0,4 ⊕ IR

(4)
0,4 = IR

+
0,4

(ii) for r odd:

IR
(1,0,1)
0,4 = IR

(1)
0,4 ⊕ IR

(3)
0,4 = IR

−
0,4

and obviously

IR
−
0,4e ◦

M
= IR−0,4e1e2e3 = IR

(0,0,2)
0,4 = IR

+
0,4.

3 The Poincaré Lemmas revisited

The Poincaré Lemmas on closed and co-closed differential forms in an open subset Ω of IRm+1 are

well-known. To our knowledge, their refined versions are less known in classical literature and so are

their statements in terms of homogeneous polynomials.

As in characterizing homogeneous polynomial solutions of (GMT)-systems of type (r, p, q), these

versions of the Poincaré Lemmas and some of their applications will play a central role (see § 4), for

convenience of the reader we restate them in full detail. Most of the proofs will be omitted since they



154 Richard Delanghe CUBO
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may be given by following classical lines of reasoning, as e.g. worked out in [3].

Throughout this section, the following notations will be kept on. For r, k ∈ IN with 0 ≤ r ≤ m + 1 and

k ≥ 1, Φrk and P
r
k denote the spaces of ∧

rIR
m+1-valued and IR

(r)
0,m+1-valued homogeneous polynomials

of degree k in IRm+1, i.e. Φrk =

P(m + 1; k) ⊗IR ∧
rIR

m+1 and Prk = P(m + 1; k) ⊗IR IR
(r)
0,m+1, where P(m + 1; k) is the space of

homogeneous real-valued homogeneous polynomials of degree k in IRm+1.

Arbitrary elements of Φrk and P
r
k will be denoted by ω

r
k and P

r
k .

Furthermore, the spaces kerrk∂
+
x and ker

r
k∂

−
x are defined by

kerrk∂
+
x =

{
P rk ∈ P

r
k : ∂

+
x P

r
k = 0

}

and

kerrk∂
−
x =

{
P rk ∈ P

r
k : ∂

−
x P

r
k = 0

}
.

3.1 The classical Poincaré Lemmas on Prk

In [3] it was shown that for each ωrk ∈ Φ
r
k,

(E ⌋d + dE ⌋)ωrk = (k + r)ω
r
k (3.1)

where the operator E ⌋ is defined by

E ⌋ =

m∑

j=0

xj ∂xj ⌋,

∂xj ⌋ being the contraction operator acting on a basic r-differential form dx
A = dxi1 ∧ dxi2 ∧ ... ∧

dxir , i.e.

∂xj ⌋dx
A =

r∑

l=1

(−1)l−1δjil dx
A\{il}

¿From (3.1) it thus follows that if dωrk = 0 with ω
r
k =

∑
|A|=r w

r
k,Adx

A, then the element ωr−1
k+1 ∈

Φr−1
k+1 given by

ω
r−1
k+1 =

1

k + r
E ⌋ωrk =

1

k + r

∑

|A|=r

ωrk,A(
r∑

j=1

(−1)j−1xij dx
A\{ij })

is such that

ωrk = dω
r−1
k+1.

Through the isomophism Θ (see § 1) we have thus proved.

Lemma 3.1 (Poincaré Lemma) Let r ≥ 1 and let k ∈ IN. Then for P rk ∈ P
r
k the following properties



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On homogeneous polynomial solutions of generalized
Moisil-Théodoresco systems in Euclidean space 155

are equivalent:

(i) ∂+x P
r
k = 0

(ii) there exists P r−1
k+1 ∈ P

r−1
k+1 such that P

r
k = ∂

+
x P

r−1
k+1 .

It is clear that by right multiplication with the pseudo-scalar eM an isomorphism is obtained be-

tween the spaces Prk and P
m+1−r
k

, its inverse being given by right multiplication with ǫM eM , where

ǫM = e
2
M = ±1. As furthermore for each P

r
k ∈ P

r
k , ∂x(P

r
k eM ) = (∂xP

r
k )eM , we have that

(∂+x P
r
k )eM = ∂

−
x (P

r
k eM )

and (3.2)

(∂−x P
r
k )eM = ∂

+
x (P

r
k eM )

Combining the relation (3.2) with Lemma 3.1 yields

Lemma 3.2 (Dual Poincaré Lemma) Let r < m + 1 and let k ∈ IN. Then for P rk ∈ P
r
k the following

properties are equivalent:

(i) ∂−x P
r
k = 0

(ii) there exists P r+1
k+1 ∈ P

r+1
k+1 such that P

r
k = ∂

−
x P

r+1
k+1 .

Refined versions of the Poincaré Lemmas may now be easily deduced.

Lemma 3.3

(i) Let r ≥ 1 and let P rk ∈ P
r
k . Then the following properties are equivalent:

(i.1) ∂+x P
r
k = 0

(i.2) there exists P r−1
k+1 ∈ P

r−1
k+1 such that ∂

−
x P

r−1
k+1 = 0 and P

r
k = ∂

+
x P

r−1
k+1 .

(ii) Let r < m + 1 and let P rk ∈ P
r
k . Then the following properties are equivalent:

(ii.1) ∂−x P
r
k = 0

(ii.2) there exists P r+1
k+1 ∈ P

r+1
k+1 such that ∂

+
x P

r+1
k+1 = 0 and P

r
k = ∂

−
x P

r+1
k+1 .

3.2 The surjectivity of the operators ∂−x ∂
+
x and ∂

+
x ∂

−
x

Let P rk ∈ ker
r
k ∂

+
x , i.e. ∂

+
x P

r
k = 0.

By means of Lemma 3.1, there exists P r−1
k+1 ∈ P

r−1
k+1 such that P

r
k = ∂

+
x P

r−1
k+1 .

As ∆x : P
r−1
k+3 → P

r−1
k+1 is surjective (see e.g. [10]) there exists P

r−1
k+3 ∈ P

r−1
k+3 such that P

r−1
k+1 =

−∆xP
r−1
k+3 .

Put P rk+2 = ∂
+
x P

r−1
k+3 . Then ∂

+
x P

r
k+2 = 0.

Moreover, on the one hand



156 Richard Delanghe CUBO
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−∆xP
r
k+2 = (∂

+
x ∂

−
x + ∂

−
x ∂

+
x )(∂

+
x P

r−1
k+3 )

= ∂+x ∂
−
x P

r
k+2

while on the other hand

−∆xP
r
k+2 = −∆x(∂

+
x P

r−1
k+3 )

= ∂+x (−∆xP
r−1
k+3 )

= ∂+x P
r−1
k+1

= P rk .

It thus follows that, given P rk ∈ ker
r
k∂

+
x , there exists P

r
k+2 ∈ ker

r
k∂

+
x such that P

r
k = ∂

+
x ∂

−
x P

r
k+2.

In a similar way, it may be proved that, given P rk ∈ ker
r
k∂

−
x , there exists P

r
k+2 ∈ ker

r
k+2∂

−
x such that

P rk = ∂
−
x ∂

+
x P

r
k+2.

We thus obtain

Lemma 3.4 The differential operators

(i) ∂−x ∂
+
x : ker

r
k+2∂

−
x → ker

r
k∂

−
x

and

(ii) ∂+x ∂
−
x : ker

r
k+2∂

+
x → ker

r
k∂

+
x

are surjective.

4 Conjugate harmonicity-Harmonic potentials-Primitives

Let the IR0,m+1-valued C1-function F defined in some open domain Ω ⊂ IR
m+1 be decomposed

following (2.3), i.e.

F = U + e0V (4.1)

where U and V are IR0,m-valued C1-functions in Ω.

Furthermore, let ∂x =
∑m

j=1 ej ∂xj be the Dirac operator in IR
m and let Dx = e0∂x = ∂x0 + e0∂x be

the Cauchy-Riemann operator in IRm+1.

Notice that the conjugate Dx of Dx is given by Dx = ∂x0 − e0∂x and that DxDx = DxDx = ∆x.

Then the following properties are equivalent in Ω:

∂xF = 0 ⇔ DxF = 0 ⇔

{
∂x0 U + ∂xV = 0

∂xU + ∂x0 V = 0
(4.2)



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On homogeneous polynomial solutions of generalized
Moisil-Théodoresco systems in Euclidean space 157

The system (4.2) clearly generalizes the classical Cauchy-Riemann system in the plane.

As left monogenic functions are real-analytic and ∂2x = −∆x, it follows that a pair (U, V ) of IR0,m-

valued C1-functions in Ω satisfying (4.2) is automatically a pair of IR0,m-valued harmonic functions

in Ω. Such a pair (U, V ) is called a conjugate harmonic pair in Ω.

4.1 Conjugate harmonicity

Let the integers r, p, q be as in section 2 with r even.

In this section we solve the following problem:

(P1) Given Uk =
∑q

j=p U
r+2j
k

, a IR
(r,p,q)
0,m -valued homogeneous harmonic polynomial of degree k in

IR
m+1, under which conditions does there exist a IR

(r−1,p,q)
0,m -valued homogeneous harmonic polynomial

Ṽk of degree k in IR
m+1 such that (Uk, Ṽk) is a conjugate harmonic pair, i.e. such that

W̃k = Uk + e0Ṽk ∈ M T
+(m + 1; k; IR

(r,p,q)
o,m+1). (4.3)

In what follows we denote by H(m + 1; k; IR
(r,p,q)
0,m ) the space of IR

(r,p,q)
0,m -valued homogeneous

harmonic polynomials of degree k in IRm+1, i.e. H(m + 1; k; IR
(r,p,q)
0,m ) = H(m + 1; k) ⊗IR IR

(r,p,q)
0,m ,

where H(m + 1; k) is the space of solid harmonics of degree k in IRm+1.

Putting Ṽk =
∑q

j=p Ṽ
r−1+2j
k

, the condition (4.3), or equivalently the condition (4.2), leads to the

following systems to be satisfied





∂−x Ṽ
r−1+2p
k

= 0

∂x0 U
r+2j
k

+ ∂+x Ṽ
r−1+2j
k

+ ∂−x Ṽ
r−1+2j+2
k

= 0, j = p, ..., q − 1

∂x0 U
r+2q
k

+ ∂+x Ṽ
r−1+2q
k

= 0

(4.4)

and





∂−x U
r+2p
k

+ ∂x0 Ṽ
r−1+2p
k

= 0

∂+x U
r+2j
k

+ ∂−x U
r+2j+2
k

+ ∂x0 Ṽ
r−1+2j+2
k

= 0

∂+x U
r+2q
k

= 0

(4.5)

¿From (4.5), it thus follows that a necessary condition on Uk to be fulfilled is that ∂
+
x U

r+2q
k

= 0.

We now claim that the latter condition is also sufficient to ensure the existence of a Ṽk conjugate

harmonic to Uk, i.e. W̃k = Uk + e0Ṽk ∈ M T
+(m + 1; k; IR

(r,p,q)
0,m+1).

The proof may be given along the same lines worked out in [2] for the construction of conjugate

harmonic pairs in general, on the understanding that, where necessary, the arguments used should be

adapted to the case of homogeneous polynomials.

That is why we do not work it out in full detail.

A first step consists in constructing

H̃k+1(x) =

∫ x0

0

Uk(t, x)dt − h̃k+1(x) (4.6)



158 Richard Delanghe CUBO
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where h̃k+1 ∈ P(m; k + 1) ⊗IR IR
(r,p,q)
0,m satisfies the equation

∆xh̃k+1(x) = ∂x0 Uk(0, x). (4.7)

As ∆x : P(m; k + 1) → P(m; k − 1) is surjective, such h̃k+1 does exist. Even more, due to the

Fischer decomposition (see [10]) the equation (4.7) admits a unique solution of the form

h̃k+1(x) = |x|
2sk−1(x)

where sk−1 ∈ P(m; k − 1) ⊗IR IR
(r,p,q)
0,m .

It may be easily verified that H̃k+1 ∈ H(m + 1; k + 1; IR
(r;p;q)
0,m ).

Fix such a solution to (4.7) and write it out as

h̃k+1(x) =

q∑

j=p

h̃
r+2j
k+1 (x).

A second step consists in proving the existence of h
r+2q
k+1 ∈ P(m; k + 1) ⊗IR IR

(r+2q)
0,m such that

W
r+2q
k+1 (x) = h̃

r+2q
k+1 + h

r+2q
k+1 (x)

satisfies the equations





∂+x W
r+2q
k+1 = 0

∂+x ∂
−
x W

r+2q
k+1 = −∂x0 U

r+2q
k

(0, x)

(4.8)

To this end first notice that, as ∂+x U
r+2q
k

= 0, we have that ∂x0 U
r+2q
k

(0, x) ∈ ker
r+2q
k−1 ∂

+
x . As

∂+x ∂
−
x : ker

r+2q
k+1 ∂

+
x → ker

r+2q
k−1 ∂

+
x is surjective (see Lemma 3.4), W

r+2q
k+1 ∈ ker

r+2q
k+1 ∂

+
x satisfying (4.8)

may be found.

Put h
r+2q
k+1 = W

r+2q
k+1 − h̃

r+2q
k+1 . Then it may be easily checked that h

r+2q
k+1 ∈

H(m; k + 1) ⊗IR IR
(r+2q)
0,m .

By construction it thus follows that

H∗k+1(x) = H̃k+1(x) − h
r+2q
k+1 (x) ∈ H(m + 1; k + 1; IR

(r,p,q)
0,m )

with (4.9)

∂+x H
∗r+2q
k+1 = 0.

Consequently W̃k = DxH
∗
k+1 = Ũk + e0Ṽk ∈ M T

+(k; IR
(r,p,q)
0,m+1) with Ũk = ∂x0 H

∗
k+1 = Uk and

Ṽk = −∂xH
∗
k+1.



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On homogeneous polynomial solutions of generalized
Moisil-Théodoresco systems in Euclidean space 159

Let H+(m + 1; k; IR
(r,p,q)
0,m ) denote the space of those elements Uk ∈ H(m + 1; k; IR

(r,p,q)
0,m ) such that

∂+x U
r+2q
k

= 0. Then we have proved

Theorem 4.1. Let Uk ∈ H(m + 1; k; IR
(r,p,q)
0,m ). Then Uk admits a conjugate harmonic Ṽk ∈

H(m + 1; k; IR
(r−1,p,q)
0,m ) if and only if Uk ∈ H+(m + 1; k; IR

(r,p,q)
0,m ).

Remarks(1) In constructing Ṽk conjugate harmonic to Uk, the harmonic potential H
∗
k+1 obtained

belongs to H+(m + 1; k + 1; IR
(r,p,q)
0,m ) (see (4.9)).

(2) Notice that the condition on Uk in Theorem 4.1 is automatically satisfied when the (GMT)-

system of type (r, p, q) considered, r being even, is such that when m + 1 is even, r + 2q = m + 1 or

when m + 1 is odd, r + 2q = m.

Indeed, let Uk =
∑q

j=p U
r+2j
k

∈ H(m + 1; k; IR
(r,p,q)
0,m ) be given.

If m + 1 = r + 2q, then clearly U
r+2q
k

= U m+1
k

= 0 and this since m < r + 2q = m + 1. Conse-

quently the condition ∂+x U
r+2q
k

= ∂+x U
m+1
k

= 0 holds.

Moreover, as we then have that ∂x0 U
m+1
k

= 0, a solution h̃k+1 to the equation (4.6) may be

chosen having its h̃m+1
k+1 -term identically zero. Finally, no correction term h

m+1
k+1 should then be taken.

If m = r + 2q, then ∂+x U
r+2q
k

= [∂xU
r+2q
k

]m+1 = 0 and so again the condition ∂
+
x U

r+2q
k

= 0 is

satisfied.

Notice that the foregoing situations are clearly met when r = 0, p = 0 and q = [ m+1
2

], i.e. when

Uk is IR
+
0,m-valued.

Remark (2) implies

Proposition 4.2 (i) Let r be even. If m+ 1 is even and r + 2q = m+ 1 or m+ 1 is odd and r + 2q = m,

then

H+(IR
m+1; k; IR

(r,p,q)
0,m ) = H(IR

m+1; k; IR
(r,p,q)
0,m )

(ii) H+(IR
m+1; k; IR+0,m) = H(IR

m+1; k; IR+0,m).

4.2 Harmonic potentials

Let again r, p, q ∈ IN be as in section 2 and let r be even.

In this section, we solve the following problem:

(P2) Let Wk ∈ M T
+(m + 1; k; IR

(r,p,q)
0,m+1) be given. Find Hk+1 ∈ H+(m + 1; k + 1; IR

(r,p,q)
0,m ) such that

Wk = DxHk+1.



160 Richard Delanghe CUBO
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To this end we first prove

Lemma 4.3 Let Wk ∈ P(m + 1; k) ⊗IR IR
(r,p,q)
0,m+1. Then the following properties are equivalent:

(i) Wk ∈ M T
+(m + 1; k; IR

(r,p,q)
0,m+1)

(ii) there exists Hk+1 ∈ H(m + 1; k + 1) ⊗IR IR
(r+1,p,q−1)
0,m+1 such that Wk = ∂xHk+1.

Proof. It is clear that if Hk+1 ∈ H(m + 1; k + 1) ⊗IR IR
(r+1,p,q−1)
0,m+1 then Wk = ∂xHk+1 ∈

P(m + 1; k) ⊗IR IR
(r,p,q)
0,m+1. As ∂

2
x = −∆x, Wk ∈ M T

+(m + 1; k; IR
(r,p,q)
0,m+1) and so (ii) → (i) is proved.

Conversely, assume that Wk ∈ M T
+(m + 1; k; IR

(r,p,q)
0,m+1) and put Wk =

∑q
j=p W

r+2j
k

. As ∂xWk = 0,

the sequence (W
r+2j
k

)
q
j=p satisfies the system (2.6), i.e.





∂−x W
r+2q
k

= 0

∂+x W
r+2j
k

+ ∂−x W
r+2(j+1)
k

= 0, j = p, ..., q − 1

∂+x W
r+2q
k

= 0

(4.10)

By virtue of Lemma 3.3, the first and last equation in (4.10) imply the existence of W
r+2p+1
k+1 ∈

P
r+2p+1
k+1 and of W

r+2q−1
k+1 ∈ P

r+2q−1
k+1 such that ∂

+
x W

r+2p+1
k+1 = 0 and W

r+2p
k

= ∂−x W
r+2p+1
k+1 , respec-

tively, ∂−x W
r+2q−1
k+1 = 0 and W

r+2q
k

= ∂+x W
r+2q−1
k+1 .

Put W ∗k =
∑q−1

j=p+1 W
r+2j
k

. Then W ∗k ∈ P(m + 1; k) ⊗IR IR
(r,p+1,q−1)
0,m+1 .

As ∆x : P(m + 1; k + 2) ⊗IR IR
(r,p+1;q−1)
0,m+1 → P(m + 1; k) ⊗IR IR

(r,p+1,q−1)
0,m+1 is surjective, there ought

to exist W ∗k+2 ∈ P(m + 1; k + 2) ⊗IR IR
(r,p+1,q−1)
0,m+1 such that ∆xW

∗
k+2 = W

∗
k .

Putting

Hk+1 = W
r+2p+1
k+1 − ∂xW

∗
k+2 + W

r+2q−1
k+1 ,

it is clear that Hk+1 ∈ P(m + 1; k + 1) ⊗IR IR
(r+1,p,q−1)
0,m+1 and that ∂xHk+1 = Wk.

Furthermore, as ∂2x = −∆x, ∂xWk = 0 implies that Wk+1 is harmonic, whence the implication (ii) →

(i) is proved.

�

Now let Hk+1 ∈ H+(m + 1; k + 1; IR
(r,p,q)
0,m ) be given. Then obviously Wk = DxHk+1 ∈ M T

+(m +

1; k; IR
(r,p,q)
0,m+1). Conversely, assume that Wk ∈ M T

+(m + 1; k; IR
(r,p,q)
0,m+1) and decompose Wk following

(4.1), i.e.

Wk = Uk + e0Vk (4.11)



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On homogeneous polynomial solutions of generalized
Moisil-Théodoresco systems in Euclidean space 161

where Uk ∈ H+(m + 1; k; IR
(r,p,q)
0,m ) and Vk ∈ H(m + 1; k; IR

(r−1,p,q)
0,m ).

Associate with Uk the harmonic potential H
∗
k+1 ∈ H+(m + 1; k + 1; IR

(r,p,q)
0,m ) obtained in (4.9), i.e.

W̃k = Ũk + e0Ṽk = DxH
∗
k+1 ∈ M T

+(m + 1; k; IR
(r,p,q)
0,m+1) with Ũk = Uk and Ṽk = −∂xH

∗
k+1.

¿From W̃k − Wk = e0(Ṽk − Vk) ∈ M T
+(m + 1; k; IR

(r,p,q)
0,m+1), i.e. Dx(W̃k − Wk) = 0, it easily follows

from (4.2) that Ṽk − Vk is independent of x0 and that Ṽk − Vk ∈ M T
+(m; k; IR

(r−1,p,q)
0,m ).

By virtue of Lemma 4.3, there exists H∗∗k+1 ∈ H(m; k + 1, IR
(r,p,q−1)
0,m ) such that Ṽk −Vk = ∂xH

∗∗
k+1.

Put

Lk+1 = H
∗
k+1 + H

∗∗
k+1.

Then by construction Lk+1 ∈ H+(m + 1; k + 1; IR
(r,p,q)
0,m ) and DxLk+1 = Wk.

Summarizing we have thus proved

Theorem 4.4 Let Wk ∈ P(m + 1; k) ⊗IR IR
(r,p,q)
0,m+1. Then the following properties are equivalent:

(i) Wk ∈ M T
+(m + 1; k; IR

(r,p,q)
0,m+1)

(ii) there exists Lk+1 ∈ H+(m + 1; k + 1; IR
(r,p,q)
0,m ) such that Wk = DxLk+1 i.e. Dx : H+(m +

1; k + 1; IR
(r,p,q)
0,m ) → M T

+(m + 1; k; IR
(r,p,q)
0,m+1) is surjective.

By virtue of Theorem 4.4 and Proposition 4.2 we obtain

Proposition 4.5(i) Let r be even. If m + 1 is even and r + 2q = m + 1 or m + 1 is odd and r + 2q = m,

then

Dx : H(IR
m+1; k + 1; IR

(r,p,q)
0,m ) −→ M T

+(IRm+1; k; IR
(r,p,q)
0,m+1)

is surjective.

(ii)Dx : H(IR
m+1; k + 1; IR+0,m) −→ M T

+(IRm+1; k; IR+0,m+1)

is surjective.

4.3 Primitives

In this section we solve the following problem:

(P.3) Let Wk ∈ M T
+(m + 1; k; IR

(r,p,q)
0,m+1) be given. Find Wk+1 ∈

M T +(m + 1; k + 1; IR
(r,p,q)
0,m+1) such that Wk = DxWk+1.

Notice that in [6] and [13] it was proved that for each Pk ∈ M
+(k), M +(k) being the space of IR0,m+1-

valued left monogenic homogeneous polynomials of degree k in IRm+1, there exists Pk+1 ∈ M
+(k + 1)

such that Pk = DxPk+1.

The specific feature in answering the problem (P.3) obviously lies in the fact that primitivation

may be realized between spaces of homogeneous polynomial solutions to a given (GMT)-system of

type (r, p, q).



162 Richard Delanghe CUBO
12, 2 (2010)

To this end, suppose that Wk ∈ M T
+(m + 1; k; IR

(r,p,q)
0,m+1).

¿From Theorem 4.2, it follows that there exists Lk+1 ∈ H+(m + 1; k + 1; IR
(r,p,q)
0,m ) such that Wk =

DxLk+1.

By means of the construction worked out in section 4.1, we may associate with Lk+1 an element L
∗
k+2 ∈

H+(m + 1; k + 2; IR
(r,p,q)
0,m ) such that W

∗
k+1 = DxL

∗
k+2 = Uk+1 + e0Vk+1 ∈ M T

+(m + 1; k + 1; IR
(r,p,q)
0,m+1)

with Uk+1 = ∂x0 L
∗
k+2 = Lk+1 and Vk+1 = −∂xL

∗
k+2.

It is then easily checked that Wk = Dx(
1
2
W ∗k+1).

We thus have proved

Theorem 4.6 Let Wk ∈ M T
+(m + 1; k; IR

(r,p,q)
0,m+1) be given. Then there exists Wk+1 ∈ M T

+(m +

1; k + 1; IR
(r,p,q)
0,m+1) such that Wk = DxWk+1.

5 The lower dimensional cases IR3 and IR4

In this section we investigate the possibilities for (GMT)-systems of type (r, p, q), r being even, in IR3

and IR4. In particular a method is described for constructing bases in the spaces M T +(4; k; IR
(r,p,q)
0,4 ).

As in each of the cases which occur, basic knowledge is used from the case M T +(3; k; IR±0,3), for conve-

nience of the reader we recall in section 5.1 some properties of the latter space. For a full description

we refer to [7].

5.1. The case M T +(3; k; IR+0,3)

Following (2.2), IR0,3 may be decomposed as

IR0,3 = IR
+
0,3 ⊕ IR

−
0,3 (5.1)

where

IR
+
0,3 = spanIR(1, e1e2, e2e0, e0e1)

and

IR
−
0,3 = spanIR(e0, e1, e2, e0e1e2).

Notice hereby that IR+0,3=̃H, the algebra of real quaternions.

As pointed out in section 2, right multiplication with the pseudo-sealar e0e1e2 establishes an isomor-

phism between IR−0,3 and IR
+
0,3, i.e.

IR
−
0,3 = IR

+
0,3e0e1e2.



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On homogeneous polynomial solutions of generalized
Moisil-Théodoresco systems in Euclidean space 163

Consequently

M T
+(3; k; IR−0,3) = M T

+(3; k; IR+0,3)e0e1e2.

In [7], it was proved that dimIRM T
+(3; k; IR+0,3) = 4(k + 1) and a method was elaborated for

constructing a basis for M T +(3; k; IR+0,3). As was also mentioned in [7], the space M T
+(3; k; IR+0,3) is

isomorphic to the space Mk(H; IR), the space of H-valued homogeneous polynomial solutions of degree

k of the Fueter operator D in IR3. A method for constructing an orthormal basis for Mk(H; IR) was

worked out in [4] and [11] (see also [12]).

Finally, let us recall that in IR3, the only (r, p, q)-subspace occuring for r even is the space IR+0,3 =

IR
(0,0,1)
0,3 .

5.2. The spaces M T +(4; k; IR
(r,p,q)
0,4 ), r even

The possible types of (GMT)-systems of type IR
(r,p,q)
0,4 have been described in section 2.

In the following subsections we shall work out methods for constructing bases for the spaces

M T +(4; k; IR
(r,p,q)
0,4 ).

5.2.1 The case IR+0,4
Let us first of all recall that if e = (e0, e1, e2, e3) is an orthogonal basis of IR

0,4 generating the Clifford

algebra IR0,4, then e = (e1, e2, e3) is an orthogonal basis of IR
0,3 which generates the Clifford algebra

IR0,3 inside IR0,4. It thus follows that

IR
+
0,4 = IR

+
0,3 ⊕ e0IR

−
0,3

where

IR
+
0,3 = spanIR(1, e2e3, e3e1, e1e2) = IR

(0,0,1)
0,3

and

IR
−
0,3 = spanIR(e1, e2, e3, e1e2e3) = IR

(1,0,1)
0,3

Notice hereby that IR+0,3=̃H = IR ⊕ V ectH where V ectH = spanIR(e2e3, e3e1, e1e2) is ismorphic

to the space of pure quaternions.

Hence, an element Wk ∈ M T
+(4; k; IR+0,4) splits into

Wk = Uk + e0Vk (5.2)

where Uk and Vk are harmonic polynomials of degree k which are, respectively, IR
+
0,3 and IR

−
0,3-

valued.



164 Richard Delanghe CUBO
12, 2 (2010)

As dimIRH(4; k) = (k + 1)
2 (see [10]), dimIRH(4; k; IR

+
0,3) = 4(k + 1)

2.

Now take a basis Sk =
{
S

(j)
k

: j = 1, ..., (k + 1)2
}

of H(4; k) and consider in

H(4; k + 1) the linearly independent subset S̃k+1 =
{
S̃

(j)
k+1 : j = 1, ..., (k + 1)

2
}

, where for each

j = 1, ..., (k + 1)2 (see also (4.6),

S̃
(j)
k+1(x) =

∫ x0

0

S
(j)
k

(t, x)dt − h̃
(j)
k+1(x) (5.3)

whith ∆xh̃
(j)
k+1(x) = ∂x0 S

(j)
k

(0, x).

Furthermore, put H̃k+1 = S̃k+1 ⊗ {1, e2e3, e3e1, e1e2}, where as usual for S ∈ S̃k+1 and

λ ∈ {1, e2e3, e3e1, e1e2} , S ⊗ λ is denoted by Sλ. Then one may easily check that DxH̃k+1 is a

linearly independent subset of M T +(4; k; IR+0,4).

Now consider again the element Wk ∈ M T
+(4; k; IR+0,4) with Wk = Uk + e0Vk (see also (5.2)) and

lift up Uk ∈ H(4; k) ⊗IR IR
+
0,3 to Ũk+1 ∈ H(4; k + 1) ⊗IR IR

+
0,3, and this following the construction

indicated above (see (5.3)).

Putting W̃k = DxŨk+1, we have that W̃k ∈ M T
+(4; k; IR+0,4) decomposes as

W̃k = Ũk + e0Ṽk (5.4)

where Ũk = ∂x0 Ũk+1 = Uk.

Consequently W̃k − Wk = e0(Ṽk − Vk) is e0IR
−
0,3-valued and left monogenic. This implies that Ṽk − Vk

is independent of x0 and that ∂x(Ṽk − Vk) = 0 or Ṽk − Vk ∈ M T
+(3; k; IR−0,3).

But M T +(3; k; IR−0,3) = M T
+(3; k; IR+0,3)e1e2e3, e1e2e3 = e ◦

M
being the pseudo-scalar of IR0,3.

Choose a basis Bk for M T
+(3; k; IR+0,3) (see secton 5.1).

Coming back to

Wk = Uk + e0Vk

we have that

Wk = W̃k + e0(Vk − Ṽk)

where W̃k ∈ spanIR(DxH̃k+1) and e0(Vk − Ṽk) ∈ spanIR(e0Bke ◦M
).

As for any a ∈ IR+0,3, e0ae ◦
M

= ae0e ◦
M

= aeM , where eM = e0e1e2e3 is the pseudo-scalar of IR0,4,

we obtain that Wk ∈ spanIR((DxH̃k+1) ∪ (BkeM )).

It may be easily verified that (DxH̃k+1) ∪ (BkeM ) is linearly independent in M T
+(4; k; IR+0,4). As

follows from the construction made above, it is also a generating set for that space, whence it is a

basis for it.



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On homogeneous polynomial solutions of generalized
Moisil-Théodoresco systems in Euclidean space 165

As H̃k+1 contains 4(k + 1)
2 elements and Bk contains 4(k + 1) elements, we have proved

Theorem 5.1

(i) dimIRM T
+(4; k; IR+0,4) = 4(k + 1)(k + 2).

(ii) ((DxH̃k+1) ∪ (BkeM )) is a basis for M T
+(4; k; IR+0,4).

Remarks(1) The space M T +(4; k; IR+0,4) coincides with the so-called space of IR
+
0,4-valued inner

spherical monogenics of degree k in IR4 (see [8]). It is well known that as a right module over the

algebra IR+0,4, dimIR+
0,4

M T +(4; k; IR+0,4) =
(k+2)(k+1)

2
.

It follows that, as dimIRIR
+
0,4 = 8, dimIRM T

+(4; k; IR+0,4) = 4(k +2)(k +1), thus confirming the result

obtained in Theorem 5.1 (i).

A classical basis for the right IR+0,4-module M
+(4; k; IR+0,4) is given by the set Bk of so-called Fueter

polynomials:

Bk =
{
Vα : α ∈ IN

3
, |α| = k

}

where for α = (α1, α2, α3),

Vα(x) =

|α|∑

j=0

(−1)j
x

j
0

j!
(e0∂x)

j (xα)

with xα = xα11 x
α2
2 x

α3
3 .

It thus follows that

B∗k =
{
Vα, Vαe0e1, Vαe0e2, Vαe0e3, Vαe1e2, Vαe1e3, Vαe2e3, Vαe0e1e2e3 : α ∈ IN

3
, |α| = k

}

is a basis for the real vector space M T +(4; k; IR+0,4).

(2) By means of Theorem 4.2 and the Remark following it, we have that Dx : H(4; k + 1; IR
+
0,3) →

M T +(4; k; IR+0,4) is surjective. As dimIRH(4; k + 1) = 4(k + 2)
2 and dimIRM T

+(4; k; IR+0,4) =

4(k + 2)(k + 1), we have that dimIR
{
Hk+1 ∈ H(4; k + 1; IR

+
0,3) : DxHk+1 = 0

}

= 4(k + 2).

5.2.2. The case IR
(2)
0,4 ⊕ IR

(4)
0,4

We apply the same procedure as worked out in the foregoing subsection.

Let Wk ∈ M T
+(4; k; IR

(2)
0,4 ⊕ IR

(4)
0,4) and write it out as

Wk = Uk + e0Vk (5.5)



166 Richard Delanghe CUBO
12, 2 (2010)

where Uk and Vk are harmonic homogeneous polynomials of degree k in IR
4 which are respec-

tively, IR
(2)
0,3 ⊕ IR

(4)
0,3 and IR

(1)
0,3 ⊕ IR

(3)
0,3 = IR

−
0,3-valued. As IR

(4)
0,3 = {0}, we have that Uk is IR

(2)
0,3-valued

with IR
(2)
0,3 = spanIR(e2e3, e3e1, e1e2)

∼= V ectH.

Starting again from a basis Sk of H(4; k), lifting it up by means of (5.3) to S̃k+1 and putting

H̃0,k+1 = S̃k+1 ⊗ {e2e3, e3e1, e1e2} we obviously have that H̃k+1 = S̃k+1 ∪ H̃0,k+1.

The reasoning made in the foregoing subsection then leads to

Theorem 5.2.

(i) (DxH̃0,k+1) ∪ (BkeM ) is a basis for M T
+(4; k; IR

(2)
0,4 ⊕ IR

(4)
0,4)

(ii) dimIRM T
+(4; k; IR

(2)
0,4 ⊕ IR

(4)
0,4) = (k + 1)(3k + 7).

Remark. As already mentioned, IR
(4)
0,3 = {0}. From Theorem 4.2 it then follows that Dx : H(4; k +

1; IR
(2)
0,3) → M T

+(4; k; IR
(2)
0,4 ⊕ IR

(4)
0,4) is surjective.

5.2.3 The case IR
(0)
0,4 ⊕ IR

(2)
0,4

Notice that IR
(0)
0,4 ⊕ IR

(2)
0,4 = (IR

(2)
0,4 ⊕ IR

(4)
0,4)eM .

We so have

Theorem 5.3

(i) (Dx(H̃0,k+1eM )) ∪ Bk is a basis for M T
+(4; k; IR

(0)
0,4 ⊕ IR

(2)
0,4)

(ii) dimIRM T
+(4; k; IR

(0)
0,4 ⊕ IR

(2)
0,4) = (k + 1)(3k + 7).

Received: March 2009. Revised: April 2009.

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