Among their many remarkable properties, Christoffel words are known to be linked to Markov numbers, i.e., positive integer solutions to the equation $x^2+y^2+z^2=3xyz$. We express Markov numbers as coefficients of Christoffel words for a rational series $S$, defined from a morphism $\mu$ mapping words to 2x2 integer matrices.
Trying to tackle the long-standing uniqueness conjecture by Frobenius, we study matrices $\mu(w)$, where $w$ is a Christoffel word, or just a word $w=aub$ with $u$ a palindrome, and consider even more general cases leading to identities for $S$. We prove some bounds on the entries of such characteristic matrices, and discuss the special cases $w=a^n b$ and $w=a b^n$ corresponding to Fibonacci and Pell numbers respectively.