Let $a,b$ are vectors in vector space $V \leq \mathcal{Cl}_n(V)$.
I would like to know if product $ababab...ab=(ab)^r$ can be written in form $\sum_{\alpha \in A} F_\alpha(a) G_\alpha(b)$. For some $F_\alpha,G_\alpha$ and some index set $A$.
So more precisely:
$(\forall r \in \mathbb{N}) \,\,(\exists \text{set} A) (\exists F_\alpha,G_\alpha : V \rightarrow \mathcal{Cl}_n(V)) (\forall a,b \in V) :abab\dots ab=(ab)^r = \sum_{\alpha \in A} F_\alpha(a) G_\alpha(b)$
Motivation: I'm reading book "geometric algebra to geometric calculus". There they stated generalized Cauchy integral formula and they said that series expansions can be obtained from it in the same way as in complex analysis. But I'm having hard times doing that.
For the start I'm trying to find expansion for $\frac{1}{x-x'}$
$$\frac{1}{x-x'} = \frac{1}{x}\frac{1}{1-x^{-1}x'} = x^{-1} \sum_{i=0}^\infty (x^{-1}x')^i$$
and now I need to write $x^{-1} \sum_{i=0}^\infty (x^{-1}x')^i$ in form $\sum_\alpha F_\alpha(x) G_\alpha(x')$ to be of any use.
In paper "Quaternionic analysis" by A. Sudbery. There is proposition 10 which says something similar for quaternions. Unfortunately proof is given only briefly and I do not understand it.