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 R^{n+1} in the Clifford algebra C_{n}, Leutwiler in Complex Variables 17 (1992) has generalized Cauchy–Riemann equations to R^{n+1} with x^{m} as one of the main solutions. But since C_{n} is not commutative, the powers x^{m} are difficult to handle. For example differentiation formulas are complicated. We present the binomial theorem in R^{n+1} which simplifies the calculation rules concerning x^{m}.

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 R^{n} 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 R^{n+1} of the Clifford algebra C_{n}.

The (universal) Clifford algebra is the associative algebra over the reals generated by the elements e_{1}, …, e_{n} satisfying the condition e_{i}e_{j} + e_{j}e_{i} = −2δ_{ij} for any i, j = 1, …, n. The vector space dimension of C_{n} is 2^{n}. 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 = x_{0} + x_{1}e_{1} + ⋯ + x_{n}e_{n}

are called vectors or paravectors. We identify this set of vectors with R^{n+1} . It is known that

(1.1)  (x_{1}e_{1} + ⋯ + x_{n}e_{n})^{2} = −∥x − x_{0}∥^{2} = −(x_{1}^{2} + … + x_{n}^{2})

1991 Mathematics Subject Classification: Primary 30G35; Secondary 11E88, 30G30.

This research is supported by the Academy of Finland.

for any vector x ∈ C_{n}. 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 C_{n} with x^{m} as one of the main solutions. But since C_{n} is not commutative, the powers x^{m} are difficult to handle. For example, differentiation formulas are complicated. Our binomial theorem in R^{n+1} simplifies the calculation rules for x^{m}.

2. The binomial theorem

Ahlfors has verified in [1] that

(z + w)^{m} = ∑_{k} \binom{m}{k}z^{k} · w^{m−k} + ρ_{m}(z, w),

where · is the Jordan product and ρ_{m} a complicated remainder. Our binomial theorem resembles the general binomial theorem

(2.1)  (a_{0} + ⋯ + a_{s})^{m} = ∑_{k_{0}+k_{1}+···+k_{s}=m} \binom{m}{k_{0}, …, k_{s}}a_{0}^{k_{0}} ⋯ a_{s}^{k_{s}},

where the generalized binomial coefficients are

\binom{m}{k_{0}, …, k_{s}} = \frac{m!}{k_{0}! ⋯ k_{s}!}.

In order to simplify our notations for a multi-index α = (α_{0}, …, α_{n}) ∈ N_{0}^{n+1} and x ∈ R^{n+1} we set

\begin{aligned}x^{α} & = x_{0}^{α_{0}} ⋯ x_{n}^{α_{n}}, \\ α! & = α_{0}! ⋯ α_{n}!, \\ |α| & = α_{0} + ⋯ + α_{n}, \\ \binom{m}{α} & = \frac{m!}{α_{0}! ⋯ α_{n}!} if |α| = m. \end{aligned}

We call a multi-index α = (α_{0}, …, α_{n}) even if all α_{0}, …, α_{n} are even. A multiindex α = (α_{0}, …, α_{n}) is also identified with a vector α_{0} +α_{1}e_{1} +⋯+α_{n}e_{n} with α_{i} ∈ N_{0} . Sometimes the notation e_{0} = 1 is convenient.

Proposition 2.1. If x is a hypercomplex number,

x^{m} = ∑_{|α|=m} \binom{m}{α}c(α)x^{α},

where the coefficients c(α) with α = (α_{0}, α_{1}, …, α_{n}) are given by

c(α) = \{ \begin{matrix}\begin{matrix}( & {\frac{1}{2}(m − α_{0}) \atop \frac{1}{2}(α − α_{0})} & ) \\ ( & {m − α_{0} \atop α − α_{0}} & ) \end{matrix}(−1)^{(m−α_{0})/2} & if α − α_{0} even, \\ \begin{matrix}( & {\frac{1}{2}(m − α_{0} − 1) \atop \frac{1}{2}(α − α_{0} − e_{i})} & ) \\ ( & {m − α_{0} \atop α − α_{0}} & ) \end{matrix}(−1)^{(m−α_{0}−1)/2}e_{i} & if α − α_{0} − α_{i}e_{i} even, \\ 0 & otherwise. \end{matrix}

Proof. Let x = x_{0} + x_{1}e_{1} + ⋯ + x_{n}e_{n}. Since x_{0} commutes with all e_{i}, we infer that

x^{m} = ∑_{α_{0}=0}^{m} \binom{m}{α_{0}}x_{0}^{α_{0}}(x_{1}e_{1} + ⋯ + x_{n}e_{n})^{m−α_{0}}.

Using (1.1) we obtain

(x_{1}e_{1} + … + x_{n}e_{n})^{s} = \{ \begin{matrix}(−1)^{s/2}(x_{1}^{2} + ⋯ + x_{n}^{2})^{s/2} if s even, \\ (−1)^{(s−1)/2}(x_{1}^{2} + ⋯ + x_{n}^{2})^{(s−1)/2}(x_{1}e_{1} + ⋯ + x_{n}e_{n}) \\ if s odd. \end{matrix}

Hence, applying (2.1) we infer that

(x_{1}e_{1} + ⋯ + x_{n}e_{n})^{m−α_{0}} = (−1)^{(m−α_{0})/2} ∑_{|ν|=(m−α_{0})/2} \binom{\frac{1}{2}(m − α_{0})}{ν}x_{1}^{2ν_{1}} ⋯ x_{n}^{2ν_{n}}

when m − α_{0} is even, and

\begin{aligned}(x_{1}e_{1} + ⋯ + x_{n}e_{n})^{m−α_{0}} & = (−1)^{(m−α_{0}−1)/2}(x_{1}e_{1} + ⋯ + x_{n}e_{n})× \\ & × ∑_{|ν|=(m−α_{0}−1)/2} \binom{\frac{1}{2}(m − α_{0} − 1)}{ν}x_{1}^{2ν_{1}} ⋯ x_{n}^{2ν_{n}} \end{aligned}

when m − α_{0} is odd. Since

\binom{m}{α_{0}} = \begin{matrix}( & {m \atop α} & ) \\ ( & {m − α_{0} \atop α − α_{0}} & ) \end{matrix},

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.

Corollary 2.2. Let x be a hypercomplex number. If m and s are natural numbers, we have

\frac{∂x^{m+s}}{∂x_{i}^{s}} = \frac{(m + s)!}{m!} ∑_{|α|=m} \binom{m}{α}c(α + se_{i})x^{α}

for any i with 0 ≤ i ≤ n.

Note that if i = 0, then c(α + se_{0}) = c(α) and ∑_{|α|=m} \binom{m}{α}c(α)x^{α} = x^{m}. Hence the usual differentiation rule ∂x^{m+s}/∂x_{0}^{s} = ((m + s)!/m!)x^{m} holds with respect to x_{0} .

Corollary 2.3. Let x be a hypercomplex number. If m is a natural number and 0 ≤ i ≤ n,

∫ x^{m}dx_{i} = \frac{1}{m + 1} ∑_{|α|=m+1} \binom{m + 1}{α}c(α − e_{i})x^{α} + g(x − x_{i}e_{i})

if c(β) = 0 for β_{i} ≤ 0 and g: R^{n+1} → C_{n} is a function.

If i = 0, we have c(α − e_{0}) = c(α) for any α with α_{0} ≥ 1. Choosing the function g as

g(x − x_{0}e_{0}) = ∑_{|α|=m+1, α_{0}=0} \binom{m + 1}{α}x^{α},

we naturally obtain ∫ x^{m}dx_{0} = x^{m+1}/(m + 1).

Corollary 2.4. Let x = x_{0} + x_{1}e_{1} + ⋯ + x_{n}e_{n} be a hypercomplex number. If m is a natural number and α = (α_{0}, …, α_{n}) a multi-index with m = |α|,

\frac{∂^{m}x^{m}}{∂xα_{0}_{0} ⋯ ∂x_{n}^{α_{n}}} = m!c(α).

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,

(x + y)^{m} = ∑_{|α|+|β|=m} \binom{m}{α, β}c(α + β)x^{α}y^{β},

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

(2.2)  (x + y)^{m} = ∑_{|γ|=m} \binom{m}{γ}c(γ)(x + y)^{γ}.

Set γ = (γ_{0}, …, γ_{n}). Substituting

(x + y)^{γ} = (x_{0} + y_{0})^{γ_{0}} ⋯ (x_{n} + y_{n})^{γ_{n}}

= ∑_{i=0,...,n^{α_{i}+β_{i}=γ_{i}}} \binom{γ_{0}}{α_{0}}\binom{γ_{1}}{α_{1}} ⋯ \binom{γ_{n}}{α_{n}}x^{α}y^{β}

= ∑_{i=0,...,n^{α_{i}+β_{i}=γ_{i}}} \frac{γ_{0}! ⋯ γ_{n}!x^{α}y^{β}}{α_{0}! ⋯ α_{n}!β_{0}! ⋯ β_{n}!}

in (2.2) we establish the assertion.

References

[1] Ahlfors, L.V.: The binomial theorem in the algebra A^{+} . - In: Constantin Carathéodory: An International Tribute, World Scientific Publ. Co, 1991, 1–15.

[2] Brackx, F., R. Delanghe, and F. Sommen: Clifford Analysis. - Pitman, Boston– London–Melbourne, 1982.

[3] Eriksson-Bique, S.-L., and H. Leutwiler: On modified quaternionic analysis in R^{3} . - Arch. Math. 70, 1998, 228–234.

[4] Gilbert, J., and M. Murray: Clifford algebras and Dirac operators in harmonic analysis. - Cambridge Stud. Adv. Math. 26, Cambridge, New York, 1991.

[5] Leutwiler, H.: Modified Clifford analysis. - Complex Variables Theory Appl. 17, 1992, 153–171.

[6] Leutwiler, H.: Modified quaternionic analysis in R^{3} . - Complex Variables Theory Appl. 20, 1992, 19–51.

[7] Leutwiler, H.: More on modified quaternionic analysis in R^{3} . - Forum Math. 7, 1995, 279–305.

[8] Leutwiler, H.: Rudiments to a function theory in R^{3} . - Expo. Math. 14, 1996, 97–123.

[9] Ryan, J. (ed.): Clifford algebras in analysis and related topics. - CRC Press, Boca Raton– New York–London–Tokyo, 1996.

[10] Sudbery, A.: Quaternionic analysis. - Math. Proc. Cambridge Philos. Soc. 85, 1979, 199– 225.

Received 29 August 1997