Let $k$ be a field of characteristic zero. Put $K=k[\![ t]\!]$ and $W=k\langle t,\partial\rangle / ([\partial,t]=1)$. Then $W$ operates on $K$ in the obvious way ($\partial f = \frac{d f}{dt}$), and define 
$$
 K^\text{fin} = \{f\in K:D f=0\text{ for some }D\in W\smallsetminus 0\} .
$$
The question is: 



> 
> Is $K^\text{fin}$ a subring of $K$?
> 
> 
> 


Apologies in advance if this question is too elementary. Assuming $K^\text{fin}$ is a ring, I would be interested if anyone knows of somewhere in which arises naturally. 



On formal solutions to differential equations
These are (more or less) called [holonomic functions](http://en.wikipedia.org/wiki/Holonomic_function). They appear in combinatorics as the generating functions of a large class of sequences (those satisfying linear recurrences with polynomial coefficients), and they're known to be closed under addition and multiplication. Here is a proof which is [due to Stanley although the result is some decades older](http://www.mathematik.uni-kassel.de/koepf/Publikationen/Algebra.pdf):


For $f \in k((t))$ let $L\_f$ denote the $k(t)$-span of $\{ f, f', f'', ..., \}$ in $k((t))$. Then $f$ is holonomic iff $L\_f$ is finite-dimensional. Since $L\_{f+g} \subseteq L\_f + L\_g$ and $L\_{fg} \subseteq L\_f L\_g$ it follows that if $L\_f$ and $L\_g$ are both finite-dimensional then so are $L\_{f+g}$ and $L\_{fg}$. Moreover $\dim L\_{f+g} \le \dim L\_f + \dim L\_g$ and $\dim L\_{fg} \le \dim L\_f \dim L\_g$. 


