Given integers \( s, t \), define a function \( \phi_{s,t} \) on the space of all formal series expansions by \(\phi_{s,t}\left(\sum a_n x^n\right) = \sum a_{sn+t} x^n.\) 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 \( (s, t) \).
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