out off the box, the FriCAS package can do the following (described in some
detail in https://arxiv.org/abs/math/0702086, the journal published a
shortened version):
1) Generating functions:
* rational (guessPade)
* algebraic (guessAlg)
* linear diffeq (guessHolo)
* polynomial diffeq (guessADE)
* Mahler type functional equations (guessFE)
* algebraic dependencies between a list of generating functions
(guessAlgDep)
2) Recurrences:
* linear, polynomial coefficients (guessPRec)
* polynomial, polynomial coefficients (guessRec)
3) Nested sums and products of the form sum_{k=1}^n f(k) and \prod_{k=1}^n
f(k), the most useful case being where f is a rational function in k. Note
that f must not depend on n, unfortunately.
4) Moreover, some special forms (guessBinRat, guessExpRat), which were not
really successful.
All of this works for lists of integers, algebraic numbers, polynomials,
and over finite fields.
Also, there are q-analogues of all these (most interesting are probably
q-differential equations).
There are some options, that can be used to restrict the search space, for
example, restricting to homogeneous differential equations, restricting the
degree of the polynomial coefficients, etc. If there is sufficient
interest, Jay Pantone found a (heuristically) better search space for
polynomial diffeqs, which would be straightforward to implement.
Martin
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