Recently, I asked WolframAlpha to give me the first few terms in the Taylor series for a complicated (but well-behaved) function. It did so, but also provided a recurrence relation for the coefficients of the series. In other words, I gave it $F(x)$ and it returned
$$F(x) = \sum_{n=0}^\infty c_nx^n,\qquad c_{n+2}=G(c_n,c_{n+1})$$
where the form of $G(c_n,c_{n+1})$, along with the values of $c_0$ and $c_1$, were also returned.
How did it do that?
I'm not familiar with a method for obtaining $G$ from $F$ and am really just looking for the name of the method or a reference.
Edit: I suppose the method of (manually evaluate the first few $c_n$) -> (notice a pattern) -> (conjecture a form for $G$) -> (prove it by induction) is one way. I'm thinking that there's a more direct way to obtain $G$ because I'm using the free version of WolframAlpha.
The assertion that the coefficients $(c_n)$ of the Taylor series satify a recurrance of the form $c_{n+2} = G(c_{n+1},c_n)$ is most likely to be the result of $F$ satisfying a linear second order differential equation:
Within the radius of convergence of the Taylor series $\sum_{n=0}^{\infty} c_n x^n$ the derivative of $F$ is given by $F'(x) = \sum_{n=0}^{\infty} nc_n x^{n-1}$, and using that fact again $F''(x) = \sum_{n=1}^{\infty} n(n-1)c_nx^{n-2}$.
Thus if $y''+a_1y'+a_0y = 0$ is a constant coefficient homogeneous 2nd order differential equation and you know that $F$ is a solution, Substituting into the differential equation then gives $$ \begin{split} 0 &= \sum_{n=1}^{\infty} c_n n(n-1)x^{n-2} + a_1\sum_{n=0}^{\infty} nc_n x^{n-1} +a_0\sum_{n=0}^{\infty} c_nx^n \\ &=\sum_{m=0}^{\infty} \left(a_0c_m + (m+1)a_1c_{m+1} +(m+2)(m+1)c_{m+2}\right)x^m \end{split} $$ Hence if follows that $$ c_{m+2} = -\frac{a_1}{m+2} c_{m+1} + \frac{a_0}{(m+1)(m+2)}c_m $$
The "Fourier transform" (informally speaking) of the above gives another way in which recurrences for the coefficients can occur: if $F(x) = \frac{p(x)}{q(x)}$ for some polynomials $p$ and $q$. In this case, rewriting the equation as $q(x)F(x)=p(x)$ and equating coefficients will give a recurrance relations for the $(c_n)$, which if $\text{deg}(q)=2$ will express $c_{n+2}$ in terms of $c_{n+1}$ and $c_n$. The polynomial $p$ will make the recurrance relations for $c_{n+2}$ depend on $n$, but only up to $\text{deg}(p)$, after which it will stabilize. In particular $F(x) = c_0.(1-a_1x^1-a_2x^2)^{-1}$ has power series $\sum_{n=0}^\infty c_nx^n$ where, if we take $c_m=0$ for $m<0$ then $c_{n+2}= a_1c_{n+1} + a_2c_n$.