We define Hecke operators U_m that sift out every m-th Taylor series
coefficient of a rational function in one variable, defined over the reals. We
prove several structure theorems concerning the eigenfunctions of these Hecke
operators, including the pleasing fact that the point spectrum of the operator
U_m is simply the set {+/- m^k, k in N} U {0}. It turns out that the
simultaneous eigenfunctions of all of the Hecke operators involve Dirichlet
characters mod L, giving rise to the result that any arithmetic function of m
that is completely multiplicative and also satisfies a linear recurrence must
be a Dirichlet character times a power of m. We also define the notions of
level and weight for rational eigenfunctions, by analogy with modular forms,
and we show the existence of some interesting finite-dimensional subspaces of
rational eigenfunctions (of fixed weight and level), whose union gives all of
the rational functions whose coefficients are quasi-polynomials.Comment: 35 pages, LaTe