I have a formal power series in one variable that I think might be algebraic (or perhaps just D-finite). Is there software that could help me explore this?


By way of comparison, there’s a very simple way to see if a formal power series appears to be rational: for small values of $n$, compute the determinant of the $(n+1)$-by-$(n+1)$ Hankel matrix whose entries are the first $2n+1$ coefficients of the formal power series. If the determinant is 0, then nontrivial elements of the nullspace correspond to possible $n$th order recurrence relations.


(I’m including the combinatorics tag since this sort of pattern-finding is sometimes an important early step in a combinatorial research project.)



Software for recognizing algebraic or D-finite formal power series
Fricas is good at that. It can be accessed via sage, once installed.



```
sage: L=[catalan_number(i) for i in range(20)]                                  
sage: fricas.guessHolo(L)                                                       
[
     n           2      ,
  [[x ]f(x): (4 x  - x)f (x) + (2 x - 1)f(x) + 1 = 0,

                     2      3      4
   f(x) = 1 + x + 2 x  + 5 x  + O(x )]
  ]
sage:  

```

and also



```
sage: fricas.guessAlg(L)                                                        
    n             2                                   2      3      4
[[[x ]f(x): x f(x)  - f(x) + 1 = 0, f(x) = 1 + x + 2 x  + 5 x  + O(x )]]

```

