Let $R$ be a field, and let $(a_n)_{n\geq 0}$ be a sequence with $a_n\in R$ for every $n\geq 0$. Suppose there exist an integer $k\geq 1$, an integer $N\geq 0$, and coefficients $c_0,c_1,\dots,c_{k-1}\in R$ such that, for every integer $n\geq N$,
be the ordinary [generating function](/page/Generating%20Function) of $(a_n)_{n\geq 0}$. Then $A(X)$ is rational: there exist polynomials $P(X),Q(X)\in R[X]$ with $Q(0)\neq 0$ such that