Fixed Points of Maps on the Space of Rational Functions

Given integers $s,t$, define a function $phi(s,t)$ on the space of all formal series expansions by $phi(s,t) (sum_n a_n x^n) = sum_n a_{sn+t}x^n$. For each function $phi(s,t)$, we determine the collection of all rational functions whose Taylor expansions at zero are fixed by $phi(s,t)$. This collection can be described as a subspace of rational functions whose basis elements correspond to certain $s$-cyclotomic cosets associated with the pair $(s, t)$.