Ralf Hemmecke <[email protected]> writes: >>> [[x^n]f(x) : -xf'(x) + f(x)^3 - f(x)^2 = 0, f(0) = 1, f'(0) = 1]. >>> >>> ??? >>> >>> Nowhere is actually explained what that notation means. :-( >> >> OK, thanks for the hint, and for discovering this inconsistency in the >> article. >> >>> Is this a common notation? Can you please give a definition for that >>> notation? >> >> Hm, I thought so, but possibly it's only notation common in >> combinatorics. > > Oh, I knew the [x^n]f(x) notation, but ... > >>> Let me guess... The sequence that is written after guessADE is obtained by >>> >>> [coefficient(f(x), n) for f in F | >>> -xf'(x) + f(x)^3 - f(x)^2 = 0 and f(0) = 1 and f'(0) = 1] >> >> well, this rather looks like we would be iterating over all f in F... > > ... you iterate over all n. n lives where exactly? Unspecified. Right?
No, I do not iterate over all n. n is a non negative integer, we are dealing with formal power series only, i.e., no Puiseux or Laurent expansions are considered. > Do you always assume that you have full information of f, i.e. f is a > given (analytic) function? (Well, analytic is too much, because you > actually deal with formal power series.) as I wrote: >> I have to admit that sometimes the equation doesn't actually determine >> the sequence, even with initial values given. (see Section 3.1, >> description of guessRec) unfortunately, rather strange things can happen, see Section 6 of the article. >>> where F is the space of all (differentiable) functions? Oh, no, F is >>> K[[x]], right? > >> Yes, for guessADE, guessHolo, guessAlg, guessPade, guessFE we consider >> the given sequence as the first few terms of a formal power series. > > But your sequence is fully given, i.e. you could compute the > coefficient up to any order n. no, although that's one would hope for. But it may happen that the equation does not determine a sequence completely. Does this answer your question? Martin -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To post to this group, send email to [email protected]. To unsubscribe from this group, send email to [email protected]. For more options, visit this group at http://groups.google.com/group/fricas-devel?hl=en.
