Some Power Series Expansions for Monogenic Functions in

Computational Methods and Function Theory
Volume 7 (2007), No. 1, 265–275
Some Power Series Expansions for Monogenic Functions
Dixan Peña Peña and Frank Sommen
(Communicated by Klaus Gürlebeck)
Abstract. New series developments for monogenic functions are presented.
The terms of these series have factors that are expressible as power functions
vanishing on special higher codimension submanifolds of Euclidean space.
These series are closely related with the Cauchy-Kowalewski extension problem
as well as to special Vekua systems arising from the consideration of axial
and biaxial symmetry.
Keywords. Clifford algebras, monogenic functions, Cauchy-Kowalewski extension
problem, Vekua systems.
2000 MSC. 30G35.
1. Introduction
Clifford analysis (see e.g. [2, 6]) offers a function theory which is a higher dimensional
analogue of the theory of the holomorphic functions of one complex
variable. The functions considered are defined in the Euclidean space R m or
R m+1 and take their values in the complex Clifford algebra C m .
Consider within this algebra the following elliptic differential operator of first
order (generalized Cauchy-Riemann operator)
m∑
∂ x0 + ∂ x = ∂ x0 + e j ∂ xj
where e 1 , . . . , e m are the generating basic elements of C m . Null solutions of this
operator are called monogenic functions (Clifford holomorphicity).
For the generalized Cauchy-Riemann operator, the computational analogy with
complex variables analysis is worked in detail by e.g. Malonek [9, 10].
Received February 2, 2006, in revised form November 8, 2006.
Published online January 19, 2007.
The first author is supported by a Doctoral Grant of the Special Research Fund of Ghent University
and the second author is supported by the FWO “Krediet aan Navorsers: 1.5.065.04”.
j=1
ISSN 1617-9447/$ 2.50 c○ 2007 Heldermann Verlag

266 D. Peña Peña and F. Sommen CMFT
We must remark that despite the fact that Clifford analysis generalizes the most
important features of classical complex analysis, the monogenic functions do not
enjoy all the properties of the holomorphic functions of one complex variable. For
instance, the product of two monogenic functions is not necessarily monogenic
because of the non-commutativity of the Clifford algebras. It is then natural to
look for techniques to construct monogenic functions.
There are a lot of techniques available to generate monogenic functions (a list of
those techniques can be found e.g. in [4]). This paper is related with two of these
techniques: the so-called axial monogenic functions and the Cauchy-Kowalewski
extension problem.
Let P k (x) be a homogeneous left monogenic polynomial of degree k in R m . The
axial monogenic functions (see [6, 12]) are monogenic functions of the form
(A(x 0 , r) + ω B(x 0 , r)) P k (x), r = |x|, rω = x,
A and B being scalar functions satisfying the Vekua-type system
∂ x0 A − ∂ r B = 2k + m − 1 B
r
∂ x0 B + ∂ r A = 0.
For more references on axial monogenic functions and special monogenic functions,
we refer to [8, 14, 15] and for the general theory of Vekua systems, we refer
to [1, 17].
The Cauchy-Kowalewski extension problem consists in finding a monogenic extension
g ∗ of a real-analytic function g defined on a given subset of R m+1 (see
e.g. [2, 3, 5, 6, 7, 11, 13, 16]).
If the given subset is an analytic m-surface in R m+1 , then in [13] Sommen proved
the existence and uniqueness of such extension. In that paper he improved the
classical Cauchy-Kowalewski extension formula obtained in [2] for functions defined
on the plane {(x 0 , x) ∈ R m+1 : x 0 = 0}. The classical Cauchy-Kowalewski
extension formula is given by the following series
∞∑
(1) g ∗ (−x 0 ) k
(x 0 , x) =
∂ k
k!
xg(x).
k=0
The Cauchy-Kowalewski extension problem has not yet been solved for surfaces
with higher codimension, except in the flat case (see [5, 6]). In this paper we
present possible methods related to solving this problem for special subsets of
R m+1 such as: spheres, cylinders and planes. Our methods also provide special
solutions of the above mentioned Vekua systems.
2. Preliminaries
Let m ∈ N and let {e 1 , . . . , e m } be an orthonormal basis of R m , then a basis
for the Clifford algebra R 0,m or its complexification C m = R 0,m ⊗ C is given by

7 (2007), No. 1 Some Power Series Expansions for Monogenic Functions 267
{e A : A ⊆ {1, . . . , m}}, where e ∅ = e 0 = 1 is the identity element, e {j} = e j ,
j = 1, . . . , m and e A = e j1 . . . e jk , A = {j 1 , . . . , j k } being ordered such that
1 ≤ j 1 < . . . < j k ≤ m. The non-commutative multiplication in the Clifford
algebra is governed by the rules:
e j e k + e k e j = −2δ jk , j, k = 1, 2, . . . , m.
Conjugation is defined as the anti-involution for which
e j = −e j , j = 1, 2, . . . , m,
with the additional rule i = −i in the case of C m .
For k = 0, 1, . . . , m fixed, we call
⎧
⎨
C (k)
m =
⎩ a ∈ C m : a = ∑
|A|=k
a A e A
⎫
⎬
⎭
the subspace of k-vectors, i.e. the space spanned by the products of k different
basis vectors. The 0-vectors and 1-vectors are simply called scalars and vectors
respectively.
If (x 0 , x 1 , . . . , x m ) ∈ R m+1 is identified with the paravector
m∑
x 0 + x = x 0 + e j x j ,
then R m+1 may be considered as a subspace of R 0,m and C m .
The product of two vectors splits up into a scalar part and a 2-vector or a socalled
bivector part:
x y = x • y + x ∧ y
where
j=1
x • y = −〈x, y〉 = −
m∑
x j y j
j=1
and
x ∧ y = ∑ e j e k (x j y k − x k y j ).
j

268 D. Peña Peña and F. Sommen CMFT
while the generalized Cauchy-Riemann operator D x = ∂ x0 +∂ x splits the Laplace
operator in R m+1 ∆ = ∂ 2 x 0
+ ∆ x = (∂ x0 + ∂ x )(∂ x0 − ∂ x ).
A continuously differentiable C m -valued function g defined in some open set of
R m (R m+1 respectively) satisfying the equation ∂ x g = 0 (D x g = 0 respectively)
is called a left monogenic function. In what follows we will say “f is monogenic”
rather than “f is left monogenic”.
Passing to spherical coordinates x = rω, r = |x| and ω ∈ S m−1 , S m−1 being the
unit sphere in R m , we have
(
∂ x = ω ∂ r + 1 )
r Γ x ,
where
Γ x = −x ∧ ∂ x = − ∑ j

7 (2007), No. 1 Some Power Series Expansions for Monogenic Functions 269
Theorem 1 (Toroidal expansion of axial type). A sufficient condition for f to
be monogenic is given by
A k,l+1 (x 0 , x) = − 1 (
∂ x0 A k,l (x 0 , x) + ω∂ r A k,l (x 0 , x)
2(l + 1)
(m − 1)ω
(3)
+ (A k,l (x 0 , x) − A l,k (x 0 , x))
2r
+ ω )
r Γ xA l,k (x 0 , x) , k, l ≥ 0.
Proof. Let z = x 0 +(r−1)i and put B k,l (x 0 , x) = Z k Z l A k,l (x 0 , x), then B k,l (x 0 , x)
may be written as
Therefore
B k,l (x 0 , x) = z
l (1 − iω)
k¯z A k,l (x 0 , x) + ¯z k l (1 + iω)
z A k,l (x 0 , x).
2
2
D x B k,l (x 0 , x) = 2lZ k Z l−1 A k,l (x 0 , x)
+Z k Z l (∂ x0 A k,l (x 0 , x) + ω∂ r A k,l (x 0 , x)
+ (m−1)ω )
A k,l (x 0 , x)
2r
(
+Z k ω
Z l r Γ xA k,l (x 0 , x) − (m−1)ω A k,l (x 0 , x)
2r
and the action of the operator D x on f is given by
∞∑ ∞∑
D x f(x 0 , x) = Z k Z
(2(l l + 1)A k,l+1 (x 0 , x) + ∂ x0 A k,l (x 0 , x)
k=0
l=0
+ ω∂ r A k,l (x 0 , x)
(m − 1)ω
+ (A k,l (x 0 , x) − A l,k (x 0 , x))
2r
+ ω )
r Γ xA l,k (x 0 , x) .
Hence we arrive at the conclusion that the recurrence relation (3) is sufficient
for f to be monogenic.
Note that the toroidal expansion (2) generates monogenic functions. All one has
to do is start from the sequence of functions {A k,0 } k≥0 (initial condition) and
calculate the functions A k,l via the recurrence formula (3).
)

270 D. Peña Peña and F. Sommen CMFT
An interesting particular case of the toroidal expansion is the case where the
coefficients do not depend on the variable x 0 and satisfy the symmetric relation
A k,l (x) = A l,k (x). With this assumption we can calculate explicitly using (3) the
coefficients of series (2). Indeed, they are determined by the formula
A k,l (x) = (−1)k+l
2 k+l k! l! ∂k+l x A 0,0 (x).
Substituting the above in (2) gives
∞∑ k∑
f(x 0 , x) = Z k−l Z l A k−l,l (x)
=
=
=
k=0
∞∑
k=0
k=0
l=0
(−1) k
2 k k!
k∑
l=0
k!
(k − l)! l! Zk−l Z l ∂ k xA 0,0 (x)
∞∑ (−1) k
2 k k! (Z + Z)k ∂xA k 0,0 (x)
∞∑ (−x 0 ) k
∂ k
k!
xA 0,0 (x)
k=0
which is the classical Cauchy-Kowalewski extension (1).
We can also solve the recurrence formula (3) if the coefficients satisfy the relation
A k,l (x 0 , x) = (−1) k+l A l,k (x 0 , x). In this case we obtain that
⎧
(−1)
⎪⎨
l
2
A k,l (x 0 , x) =
k+l k! l! D x(P x D x ) (k+l−1)/2 A 0,0 (x 0 , x) if k + l is odd,
⎪⎩ (−1) l
2 k+l k! l! (P xD x ) (k+l)/2 A 0,0 (x 0 , x) if k + l is even,
where the differential operator P x is defined by
P x g = ∂ x0 g + ω∂ r g + 1 r Γ x(ωg).
Then we obtain the following expansion
∞∑ 1
f(x 0 , x) =
2 2k (2k)! (Z − Z)2k (P x D x ) k A 0,0 (x 0 , x)
k=0
∞∑ 1
+
2 2k+1 (2k + 1)! (Z − Z)2k+1 D x (P x D x ) k A 0,0 (x 0 , x)
k=0
∞∑
k (r − 1)2k
= (−1) (P x D x ) k A 0,0 (x 0 , x)
(2k)!
k=0
∞∑
+ (−1) k (r − 1)2k+1 ω
D x (P x D x ) k A 0,0 (x 0 , x).
(2k + 1)!
k=0

7 (2007), No. 1 Some Power Series Expansions for Monogenic Functions 271
This expansion may be considered as a kind of Cauchy-Kowalewski extension for
the cylinder r = 1.
Now we investigate the generalization of the previous theorem to the biaxially
symmetric case. To that end we split up R m as R m = R p 1
⊕ R p 2
, p 1 + p 2 = m,
yielding
p 1
p
∑
∑ 2
x = x 1 + x 2 , x 1 = e j x j , x 2 =
and accordingly
j=1
j=1
e p1 +jx p1 +j
p 1
p
∑
∑ 2
∂ x = ∂ x1 + ∂ x2 , ∂ x1 = e j ∂ xj , ∂ x2 = e p1 +j∂ . xp1 +j
Introducing spherical coordinates on R p 1
and R p 2
respectively, i.e.
we thus have that
where
j=1
j=1
x 1 = r 1 ω 1 , r 1 = |x 1 |, ω 1 ∈ S p 1−1
x 2 = r 2 ω 2 , r 2 = |x 2 |, ω 2 ∈ S p 2−1
∂ x = ω 1
(
∂ r1 + 1 r 1
Γ x1
)
+ ω 2
(
∂ r2 + 1 r 2
Γ x2
)
Γ x1 = −x 1 ∧ ∂ x1 and Γ x2 = −x 2 ∧ ∂ x2 .
We call a toroidal expansion of biaxial type a convergent series of biaxial type
around S p1−1 × S p2−1 of the form
∞∑ ∞∑
f(x 1 , x 2 ) = Z k Z l A k,l (x 1 , x 2 ), Z = (r 1 − 1) + (r 2 − 1)ω 1 ω 2 .
k=0
l=0
We thus obtain the following generalization of Theorem 1.
Theorem 2 (Toroidal expansion of biaxial type). A sufficient condition for f
to be monogenic is given by
(
ω
A k+1,l (x 1 , x 2 ) = 1
(ω
2(k + 1) 1 ∂ r1 A k,l (x 1 , x 2 ) + 1 )
Γ x1 A l,k (x
r 1 , x 2 )
1
+ω 2
(∂ r2 A k,l (x 1 , x 2 ) + 1 )
Γ x2 A l,k (x
r 1 , x 2 )
(4)
( 2
(p1 − 1)ω
+
1
+ (p )
2 − 1)ω 2
2r 1 2r 2 )
× (A k,l (x 1 , x 2 ) − A l,k (x 1 , x 2 )) , k, l ≥ 0.

272 D. Peña Peña and F. Sommen CMFT
Proof. Let z = (r 1 − 1) + (r 2 − 1)i and put B k,l (x 1 , x 2 ) = Z k Z l A k,l (x 1 , x 2 ), then
we have that
B k,l (x 1 , x 2 ) = z k¯z l (1 − iω 1ω 2 )
2
Therefore
∂ x B k,l (x 1 , x 2 ) = 2kZ k−1 Z l ω 1 A k,l (x 1 , x 2 )
A k,l (x 1 , x 2 ) + ¯z k z l (1 + iω 1ω 2 )
A k,l (x
2
1 , x 2 ).
+Z k Z l (ω 1 ∂ r1 A k,l (x 1 , x 2 ) + ω 2 ∂ r2 A k,l (x 1 , x 2 )
( (p1 − 1)ω
+
1
+ (p ) )
2 − 1)ω 2
A k,l (x
2r 1 2r 1 , x 2 )
2
(
+Z k Z l ω 1
Γ x1 A k,l (x
r 1 , x 2 ) + ω 2
Γ x2 A k,l (x
1 r 1 , x 2 )
2
( (p1 − 1)ω
−
1
+ (p ) )
2 − 1)ω 2
A k,l (x
2r 1 2r 1 , x 2 )
2
and the action of the operator ∂ x on f is given by
∂ x f(x 1 , x 2 ) =
∞∑
k=0
∞∑
Z k Z
(2(k l + 1)ω 1 A k+1,l (x 1 , x 2 )
l=0
+ω 1
(∂ r1 A k,l (x 1 , x 2 ) + 1 )
Γ x1 A l,k (x
r 1 , x 2 )
1
+ω 2
(∂ r2 A k,l (x 1 , x 2 ) + 1 )
Γ x2 A l,k (x
r 1 , x 2 )
2
( (p1 − 1)ω
+
1
+ (p )
2 − 1)ω 2
2r 1 2r 2
)
× (A k,l (x 1 , x 2 ) − A l,k (x 1 , x 2 ))
.
We thus have that the recurrence relation (4) is sufficient for function f to be
monogenic.
Note that for the toroidal expansion of biaxial type the sequence of functions
{A 0,l (x 1 , x 2 )} l≥0 is the initial condition.

7 (2007), No. 1 Some Power Series Expansions for Monogenic Functions 273
Let P x1 and P x2 be the differential operators defined by
P x1 g = ω 1 ∂ r1 g + 1 r 1
Γ x1 (ω 1 g),
P x2 g = ω 2 ∂ r2 g + 1 r 2
Γ x2 (ω 2 g).
In a completely analogous manner as in the axial case, using (4) we obtain the following
Cauchy-Kowalewski like extensions around S p 1−1 and S p 2−1 respectively.
(i) A k,l (x 1 , x 2 ) = A l,k (x 1 , x 2 ):
∞∑
f(x 1 , x 2 ) = (−1) k (r 1 − 1) 2k ( ) k
(Px1 − ∂ x2 )∂ x A0,0 (x
(2k)!
1 , x 2 )
k=0
∞∑
+ (−1) k (r 1 − 1) 2k+1 ω 1
( ) k
∂ x (Px1 − ∂ x2 )∂ x A0,0 (x
(2k + 1)!
1 , x 2 )
k=0
(ii) A k,l (x 1 , x 2 ) = (−1) k+l A l,k (x 1 , x 2 ):
∞∑ (r 2 − 1) 2k ( ) k
f(x 1 , x 2 ) =
(∂x1 − P x2 )∂ x A0,0 (x
(2k)!
1 , x 2 )
k=0
∞∑ (r 2 − 1) 2k+1 ω
+
2
( ) k
∂ x (∂x1 − P x2 )∂ x A0,0 (x
(2k + 1)!
1 , x 2 ).
k=0
4. Another special expansion
Motivated by both the Cauchy-Kowalewski extension and the Fischer decomposition
(see [6]) we consider the series
∞∑ ∞∑
f(x 0 , x 1 , x 2 ) = (x 0 + x 1 ) k (x 0 − x 1 ) l A k,l (x 0 , x 1 , x 2 ).
k=0
l=0
We are thus led to the following theorem.
Theorem 3. Sufficient for f to be monogenic is the recurrence relation
A k,l+1 (x 0 , x 1 , x 2 ) = − 1 (
∂ x0 A k,l (x 0 , x
2(l + 1)
1 , x 2 ) + ω 1 ∂ r1 A k,l (x 0 , x 1 , x 2 )
(5)
+ (p 1 − 1)ω 1
2r 1
× (A k,l (x 0 , x 1 , x 2 ) − A l,k (x 0 , x 1 , x 2 ))
+ ω 1
Γ x1 A l,k (x 0 , x
r 1 , x 2 )
1 )
+∂ x2 A l,k (x 0 , x 1 , x 2 ) , k, l ≥ 0.

274 D. Peña Peña and F. Sommen CMFT
Proof. The proof is similar to the proof of Theorem 1.
Let p 1 = 1. For this particular case we can assume that the coefficients in the
above series only depend on the variable x 2 (note that x 2 = ∑ m
j=2 x je j ). Then
the recurrence relation (5) takes the form
A k,l+1 (x 2 ) = − 1
2(l + 1) ∂ x 2
A l,k (x 2 ), k, l ≥ 0.
Solving this recurrence relation we get that
⎧
l (k − l)!
⎪⎨ (−1)
4
A k,l (x 2 ) =
l k! l! ∆l x 2
A k−l,0 (x 2 ) if k ≥ l,
⎪⎩ (−1) k+1 (l − k − 1)!
∂
2 4 k x2 ∆ k x
k! l!
2
A l−k−1,0 (x 2 ) if k < l,
where ∆ x2 = ∑ m
j=2 ∂2 x j
.
We have thus obtained the following codimension 2 generalization of the Cauchy-
Kowalewski Extension Theorem.
Corollary 1. Let A k,0 (x 2 ) (k ≥ 0) be given functions, and consider the formal
series
∞∑ ∞∑
m∑
f(x 0 , x 1 , x 2 ) = (x 0 + e 1 x 1 ) k (x 0 − e 1 x 1 ) l A k,l (x 2 ), x 2 = x j e j .
k=0
l=0
Then there exist unique functions A k,l (x 2 ), k ≥ 0, l > 0, such that the above
sum f is monogenic. Moreover, those functions can be calculated using the above
formula.
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Dixan Peña Peña
E-mail: dixan@cage.UGent.be
Address: Ghent University, Department of Mathematical Analysis, Galglaan 2, B-9000 Gent,
Belgium.
Frank Sommen
E-mail: fs@cage.UGent.be
Address: Ghent University, Department of Mathematical Analysis, Galglaan 2, B-9000 Gent,
Belgium.