Computes a closed-form formula for the power series representation of
f in the variable z and the summation index n. The output is either a
hypergeometric type series, or a recursive formula deduce from the
recurrence equation satisfied by series coefficients.
Note that no prefix is needed for this procedure even when the package is not loaded (with(FPS)) in Maple.
FPS(f,z,n,z0): for expansion about z0.
By default, the procedure searches for a hypergeometric type series
representation. If this is not found but a holonomic recurrence equation
instead, then a recursive formula is given from that recurrence
relation. Otherwise, a third approach base on computing quadratic
differential equations is used.
FPS(f,z,n,onlyhyper=boolean): if boolean is true, then the computations stop once the search for a hypergeometric type power series is completed.
FPS(f,z,n,hypergeomcoeffs=true): to search for a hypergeometric type representation where every m-fold hypergeometric term solution of the underlying recurrence equation appears in a different summation.
FPS(f,z,n,fpstype=name): name can be 'holonomic', 'quadratic' or 'specialfunctions'.
holonomic: for deducing a recursive formula of the series from a holonomic recurrence equation. In Maxima, this is done using HoloRep(f,z,n).
quadratic: for deducing a recursive formula of the series from a quadratic recurrence equation. In Maxima, this is done by the QNF(f,z,n).
specialfunctions: to do the same computations when special functions appear in the input. This is not a proper optional method. The version incorporated in Maple does not need such a specification from the user since special functions can be detected by internal checking.
FPS(f,z,n,maxdeorder=M): To allow the search for a higher-order differential equation in order to have more chance in finding a representation of hypergeometric type or a holonomic recursive formula. This is done in Maxima by increasing the value of Nmax.
The Maxima package applied further techniques (Cauchy product, reciprocal of a series) to cover some other cases.