  Given integers s and t, define a function phi_{s,t} on the space of all
formal complex series expansions by phi_{s,t} (sum a_n x^n) = sum a_{sn+t} x^n.
We define an integer r to be distinguished with respect to (s,t) if r and s are
relatively prime and and r divides t (1 + s + ... s^{ord_r(s)-1}). The vector
space consisting of all rational functions whose Taylor expansions at zero are
fixed by phi_{s,t} was previously classified by constructing a basis that is
partially indexed by integers that are distinguished with respect to the pair
(s,t). In this paper, we study the properties of the set of distinguished
integers with respect to (s,t). In particular, we demonstrate that the set of
distinguished integers with respect to (s,t) can be written as a union of
infinitely many arithmetic progressions. In addition, we construct another
generating set for the collection of rational functions that are fixed by
phi_{s,t} and discuss the relationship between this generating set and the
basis that was generated previously.
