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!