I have a proof of the following fact related to ordinary generating functions, and I was curious if it was known, as it seems plausible it is classically known:

“Let $lambda_1,ldots, lambda_k$ be algebraic numbers. Let $f(z)= sum^infty_{n=0} c_nz^n$ when each $c_nin mathbb{Z}.$ Suppose $f$ analytically continues to $mathbb{C}setminus {lambda_1,ldots,lambda_k}.$ Then, $f$ is a rational function.”