Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 24, 1999, 225–229
THE BINOMIAL THEOREM
FOR HYPERCOMPLEX NUMBERS
Sirkka-Liisa Eriksson-Bique
University of Joensuu, Department of Mathematics
P.O. Box 111, FIN-80101 Joensuu, Finland; Sirkka-Liisa.Eriksson-Bique@joensuu.fi
Abstract. The theory of complex variables is based on considerations of z m . Imbedding Rn+1 in the Clifford algebra Cn , Leutwiler in Complex Variables 17 (1992) has generalized
Cauchy–Riemann equations to Rn+1 with xm as one of the main solutions. But since Cn is
not commutative, the powers xm are difficult to handle. For example differentiation formulas are
complicated. We present the binomial theorem in Rn+1 which simplifies the calculation rules
concerning xm .
1. Introduction
Several attempts have been made to generalize one-variable complex analysis
to higher dimensions. The starting point is to consider the Euclidean space Rn as
a subspace of some algebra over the reals. Naturally, one would like the possibility
of division in this algebra. The Frobenius theorem states that an associative
algebraic division algebra over the reals is always isomorphic to either R , the field
of complex numbers C, or the division algebra of real quaternions H . Hamilton
discovered quaternions in 1866. The algebra of quaternions is not commutative and
therefore the usual binomial theorem fails there. We present a binomial theorem
in H and even in a more general subspace Rn+1 of the Clifford algebra Cn .
The (universal) Clifford algebra is the associative algebra over the reals generated by the elements e1 , . . . , en satisfying the condition ei ej + ej ei = −2δij for
any i, j = 1, . . . , n . The vector space dimension of Cn is 2n . When n = 1 , we
obtain the complex number system. When n = 2 , we obtain the set of quaternions. Clifford algebras provide a rich framework for generalizing many results
from one-variable complex analysis ([2], [4], [10], [9]).
Clifford numbers of the form
x = x0 + x1 e1 + · · · + xn en
are called vectors or paravectors. We identify this set of vectors with Rn+1 . It is
known that
(1.1)
(x1 e1 + · · · + xn en )2 = −x − x0 2 = −(x21 + . . . + x2n )
1991 Mathematics Subject Classification: Primary 30G35; Secondary 11E88, 30G30.
This research is supported by the Academy of Finland.
226
Sirkka-Liisa Eriksson-Bique
for any vector x ∈ Cn . For this reason vectors are also called hypercomplex
numbers.
The theory of complex variables is based on considerations of z m . Leutwiler
has in [5], [6], [7] and [8] generalized the Cauchy–Riemann equations to Cn with
xm as one of the main solutions. But since Cn is not commutative, the powers
xm are difficult to handle. For example, differentiation formulas are complicated.
Our binomial theorem in Rn+1 simplifies the calculation rules for xm .
2. The binomial theorem
Ahlfors has verified in [1] that
m
(z + w)
=
m
k
k
z k · wm−k + ρm (z, w),
where · is the Jordan product and ρm a complicated remainder. Our binomial
theorem resembles the general binomial theorem
m
m
ak0 · · · aks s ,
(2.1)
(a0 + · · · + as ) =
k0 , . . . , ks 0
k0 +k1 +···+ks =m
where the generalized binomial coefficients are
m!
m
=
.
k0 ! · · · ks !
k0 , . . . , ks
In order to simplify our notations for a multi-index α = (α0 , . . . , αn ) ∈ Nn+1
0
and x ∈ Rn+1 we set
αn
0
xα = xα
0 · · · xn ,
α! = α0 ! · · · αn !,
|α| = α0 + · · · + αn ,
m
m!
if |α| = m.
=
α0 ! · · · αn !
α
We call a multi-index α = (α0 , . . . , αn ) even if all α0 , . . . , αn are even. A multiindex α = (α0 , . . . , αn ) is also identified with a vector α0 + α1 e1 + · · · + αn en with
αi ∈ N0 . Sometimes the notation e0 = 1 is convenient.
Proposition 2.1. If x is a hypercomplex number,
m
m
x =
c(α)xα ,
α
|α|=m
The binomial theorem for hypercomplex numbers
227
where the coefficients c(α) with α = (α0 , α1 , . . . , αn ) are given by
 1

2 (m − α0 )



1

)

2 (α − α0

(−1)(m−α0 )/2
if α − α0 even,



m
−
α
0




α − α0
c(α) = 1 (m − α0 − 1) 2



1

− α0 − ei )

2 (α


(−1)(m−α0 −1)/2 ei if α − α0 − αi ei even,


m − α0




α − α0

0
otherwise.
Proof. Let x = x0 + x1 e1 + · · · + xn en . Since x0 commutes with all ei , we
infer that
m m α0
m
x0 (x1 e1 + · · · + xn en )m−α0 .
x =
α
0
α =0
0
Using (1.1) we obtain
s
(x1 e1 + . . . + xn en ) =
(−1)s/2 (x21 + · · · + x2n )s/2 if s even,
(−1)(s−1)/2 (x21 + · · · + x2n )(s−1)/2 (x1 e1 + · · · + xn en )
if s odd.
Hence, applying (2.1) we infer that
(x1 e1 + · · · + xn en )
m−α0
(m−α0 )/2
= (−1)
|ν|=(m−α0 )/2
1
2 (m
− α0 ) 2ν1
n
x1 · · · x2ν
n
ν
when m − α0 is even, and
(x1 e1 + · · · + xn en )m−α0 = (−1)(m−α0 −1)/2 (x1 e1 + · · · + xn en )×
1
(m − α0 − 1) 2ν1
2
n
x1 · · · x2ν
×
n
ν
|ν|=(m−α0 −1)/2
when m − α0 is odd. Since
m
m
α
,
= m − α0
α0
α − α0
the result follows.
For n = 2 the preceding theorem is presented in a slightly different form in [3,
p. 233]. Using the preceding theorem it is easy to differentiate or integrate powers
of x.
228
Sirkka-Liisa Eriksson-Bique
Corollary 2.2. Let x be a hypercomplex number. If m and s are natural
numbers, we have
∂xm+s
(m + s)! m
=
c(α + sei )xα
∂xsi
m!
α
|α|=m
for any i with 0 ≤ i ≤ n .
m
α
m
Note that if i = 0 , then c(α + se0 ) = c(α) and
|α|=m
α c(α)x = x .
Hence the usual differentiation rule ∂xm+s /∂xs0 = (m + s)!/m! xm holds with
respect to x0 .
Corollary 2.3. Let x be a hypercomplex number. If m is a natural number
and 0 ≤ i ≤ n ,
1
x dxi =
m+1
m
|α|=m+1
m+1
c(α − ei )xα + g(x − xi ei )
α
if c(β) = 0 for βi ≤ 0 and g: Rn+1 → Cn is a function.
If i = 0 , we have c(α − e0 ) = c(α) for any α with α0 ≥ 1 . Choosing the
function g as
m+1 α
x ,
g(x − x0 e0 ) =
α
|α|=m+1, α0 =0
we naturally obtain
xm dx0 = xm+1 /(m + 1).
Corollary 2.4. Let x = x0 + x1 e1 + · · · + xn en be a hypercomplex number.
If m is a natural number and α = (α0 , . . . , αn ) a multi-index with m = |α| ,
∂ m xm
= m!c(α).
n
∂x0α0 · · · ∂xα
n
The following binomial theorem for hypercomplex numbers is obtained.
Theorem 2.5. Let x and y be hypercomplex numbers. If m is a natural
number,
m m
c(α + β)xα y β ,
(x + y) =
α, β
|α|+|β|=m
where the coefficients c( · ) are the same as in Proposition 2.1.
The binomial theorem for hypercomplex numbers
229
Proof. Using Proposition 2.1 we infer
m
(2.2)
(x + y)
m
=
c(γ)(x + y)γ .
γ
|γ|=m
Set γ = (γ0 , . . . , γn ). Substituting
(x + y)γ = (x0 + y0 )γ0 · · · (xn + yn )γn
γ0 γ1 γn α β
=
···
x y
α
α
α
0
1
n
α +β =γ
i
i
i
i=0,...,n
=
αi +βi =γi
γ0 ! · · · γn !xα y β
α0 ! · · · αn !β0 ! · · · βn !
i=0,...,n
in (2.2) we establish the assertion.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
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London–Melbourne, 1982.
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225.
Received 29 August 1997