An application of linear species (Q1336677)

Using linear species, the authors prove the following generalization of the cyclotomic identity \[ {1\over 1-\alpha x} = 1 + \sum _{k\geq 1} S (\alpha,k) {x ^ k \over 1-x^ k}, \] where \( S (\alpha, k)\) denotes the number of aperiodic words of length \(k\) over an alphabet with \(\alpha\) letters.