Recently I am studying generating function with the polynomial coefficient. So, for example, $a_n=x^n$ and the generating function is $F(x,y)=\sum a_n y^n$. Is this a rational generating function or a holonomic generating function? Is there are a method to show it?
The polynomial ring $C[x]$ has two more interesting basis $x^{\underline{n}}:=x(x-1)\cdots(x-(n-1))$ and $x^{\overline{n}}:=x(x+1)\cdots(x+n-1)$
How can I show the generating function of $F'(x,y)=\sum x^{\underline{n}}y^n$ is holonomic? Same question for $F'(x,y)=\sum x^{\overline{n}}y^n$.
For the first case, take advantage of the closed form:
$$F(x,y) = \sum x^n y^n = \frac{1}{1 - xy}.$$
For the other two, use the fact that $x^{\underline{n}}$ is the generating function of the Stirling numbers of the second kind and use recurrence relations for them; for $x^{\overline{n}}$ is the generating function of the Stirling numbers of the first kind and use recurrence relations similarly.