# abstract algebra – Generalization of the geometric series to multiple variables using Lyndon words

The following claim can be found, without proof, in A Formula for the Determinant of a Sum of Matrices by Reutenauer and Schützenberger, (I am paraphrasing):

In the algebra of noncommutative power series with indeterminates from a set $$A$$ with integer coefficients, we have,
$$(1 – a_1 – ldots – a_k)^{-1} = prod_{l} (1 – l)^{-1},$$
where the product is taken over all Lyndon words in decreasing order.

Here, $$A$$ is a finite totally ordered set $$A={a_1, ldots, a_k}$$ satisfying $$a_1 < ldots < a_k$$, whereas $$l$$ stands for a Lyndon word, and the product is taken over all Lyndon words. In particular, the right-hand side is an infinite product.

My question is: how can one prove this fact?

It is easy to see that when $$A$$ is a singleton set $$A = {a_1}$$, the fact above reduces to the geometric series. However, I do not see how to generalize the geometric series to multiple variables.

Any help is appreciated!