Recently I read on wiki (see here): "Any periodic sequence can be constructed by element-wise addition, subtraction, multiplication and division of periodic sequences consisting of zeros and ones."
On the other hand it is commonly known that such a "periodic sequences consisting of zeros and ones" correspond to a class of Generating Function of the type:
$$G(x)=\frac{x^{\gamma-1}}{1-x^\gamma} \qquad(1)$$
while $\gamma$ denotes the period from a $1$ to the next $1$ in the sequence.
Question: is there a theory whether the above claim can be also applied to the Generating Functions of the sequences, in other words, is there a way to deduct the Generating function of any arbitrary periodic sequence from the above class of Generating functions $(1)$ (by addition, subtraction, multiplication and division or else)?