**Added.** (9/8/2013). Here is an explicit case. Recall that the series $\sum_{n \geq 0} \binom{2n}{n}t^n = (1-4t)^{-1/2}$ is algebraic. Certainly there are polynomials $f(n)$, all of which split into linear factors over $\mathbb{Q}$, such that the subring $\mathbb{Z} \big[ \binom{2n}{n}/f(n) \big] \subset \mathbb{Q}$ is finitely generated; for example, $f(n) = n+1$ produces the Catalan numbers. Can it be shown that there are no other examples of constant coefficients linear recurrence sequences $f(n)$, alias confluent power sums, which have this property? (If one prefers, that $f(n)$ divides $\binom{2n}{n}$ for each $n$.) My answer below implies that there are none under the added assumption that the rational function $\sum_{n \geq 0} f(n)t^n$ has a dominant pole at some place.

It must surely be true that "most" power sums contain many values having a large prime factor ("most" here means outside of degenerate cases), but I don't think enough is known here unconditionally to directly yield the positive answer to the question above.

**Original post.** The following well-known theorem has been proved by Y. Pourchet and A. van der Poorten, and strengthened by P. Corvaja and U. Zannier. Let $K$ be a field of characteristic zero and $f := \sum_{n \geq 0} a(n)t^n, \, g:= \sum_{n \geq 0} b(n)t^n \in K[[t]]$ the power series expansions of two rational fractions, where $b(n)$ are assumed to be non-zero. If the coefficients of the "Hadamard quotient" $h := \sum_{n \geq 0} a(n)/b(n) t^n$ are contained in a finitely generated ring, then $h$ is also rational. Both proofs are specific to number fields (to which the characteristic zero case boils down after an easy reduction): the former (by Pourchet and van der Poorten) is by a $p$-adic extrapolation, and makes essential use of the mixed characteristic of $\mathbb{Q}_p$; the later (by Corvaja and Zannier) is based on diophantine approximations (Schmidt's Subspace theorem).

Has the positive characteristic ("function field") variant been considered anywhere? How about the following generalizations?

If the Hadamard quotient of an algebraic by a rational functions has all its coefficients drawn from a finitely generated ring, then it is itself an algebraic function.

The same for the Hadamard quotient of a $D$-finite by a rational functions.

A variant for $G$-functions, where I restrict for simplicity to the number field case. Define the height of the power series $f$ above to be $\limsup_n \frac{1}{n} h(f_{/n})$, where $f_{/n} := \sum_{j=0}^n a(j) t^j$ is the polynomial truncation modulo $t^{n+1}$, and $h(P)$ is the projective logarithmic absolute height of the set of coefficients of a non-zero polynomial in $\bar{\mathbb{Q}}[t]$. A "$G$-function" is a power series in $\bar{\mathbb{Q}}[[t]]$ which is $D$-finite and has finite height. Such power series are closed under the Hadamard product, and one may ask:

*If the Hadamard quotient of two $G$-functions has finite height, is it $D$-finite*?A variant for diagonals of rational functions in several variables (those again are stable under Hadamard product): if the Hadamard quotient of two such functions has all its coefficients drawn from a finitely generated ring, is it itself the diagonal of a rational function?

Finally, a variant for $G$-functions of motivic origin, i.e. satisfying a Picard-Fuchs differential equation for the variation of cohomology of a family of algebraic varieties. The class of such power series is again stable under Hadamard product; a beautiful conjecture of Bombieri and Dwork suggests that perhaps all $G$-functions have a motivic origin. One may ask: if the Hadamard quotient of two motivic $G$-functions has a finite height, does it also satisfy a Picard-Fuchs equation?