When does a power series converge to a rational function?

Question

Are there any results to determine whether the given power series of real variable converges to a rational function? I mean just analyzing the coefficients of the series. One way is to find the sum function which is not always easy to find.

Answer 1

A power series is the Taylor series of a rational function if and only if its terms satisfy a constant-coefficient linear recurrence $$ a_n = \sum_{j=1}^m c_j a_{n-j}$$ for all sufficiently large $n$, where $m$ and $c_j$ are constants.

EDIT: This is certainly well known, and easier to prove than to find a printed reference. If $R(x) = A(x)/B(x)$ with $A(x)$ and $B(x)$ polynomials, then $B(x) R(x) = A(x)$ gives you the recurrence $\sum_i b_i r_{k-i} = 0$ for $k > \text{degree}(A)$, where $b_i$ are the coefficients of $B(x)$ and $r_i$ the Maclaurin coefficients of $R(x)$. Conversely, if $\sum_{j=0}^m c_j a_{k-i} = 0$ for $k > n$, that says $B(x) G(x)$ is a polynomial of degree $\le n$ where $G(x)$ is the generating function $\sum_k a_k x^k$ and $B(x) = \sum_{j=0}^m c_j x^j$.