GFUN (1231641)

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###################### INTRODUCTION ################################ The Gfun package has been designed as an help for the manipulation and discovery of generating functions. Given the first terms of the sequence, the package contains functions that will help conjecture what the generating function is. In some cases, this answer will be ``explicit''. In most cases though, such an explicit expression will not exist, and the answer will be an equation (either differential or algebraic) satisfied by the generating function. The Gfun package also provides tools to compute with generating functions defined by equations. For instance, given two generating functions defined by linear differential equations with polynomial coefficients, there is a procedure to compute the differential equation satisfied by their product. #################### WHAT'S NEW ################################# compared to the version of gfun that was in Maple's share library. - Almost everything has been rewritten. In particular, attention has been paid to consistency both in the working of the package and in the naming of its procedures. Besides, most of the functions should be faster. ### Correctness: ##### - The initial conditions are now handled (more) correctly. - Non-homogeneous equations are now handled (more) correctly. ### Consistency: ##### - In the guessing functions, the first term of the list is now considered as the 0th term of the sequence instead of the first one. This makes it possible to use series or lists without any difference. - exptoord and ordtoexp are now called borel and invborel - hsum is now diffeq+diffeq - hproduct is now diffeq*diffeq - HadamardProduct is now hadamardproduct - algfun has been replaced by algeq everywhere, which is more consistent since diffeq is used for differential equations. ### Additions: ##### - The reverse of listtoseries(l,x,egf) now exists, it's called listtoseries(l,x,laplace). - A new function guesseqn has been added. It does the same thing as guessgf, except that it does not try to solve the differential equation. - New functions rec+rec and rec*rec have been added. - A variable gfun[version] has been added. Its value is 2.0. - A function algebraicsubs has been added. It takes as input a LDE for f(z), a polynomial P(y,z) defining an algebraic function y(z), and yields a LDE for f(y(z)). ### Misc. changes and improvements: ##### - Default type of series in listtoseries is now the first element of optionsgf (i.e. by default ogf) instead of egf previously. - ratpolytocoeff now uses fullparfrac from the share library and thus never computes any factorisation. - algeqtodiffeq should be much faster. - rectoproc now produces several types of procedures: _ if remember is specified as the third argument, then the procedure which is produced will use option remember; it will then compute u(n) in linear number of arithmetic operations and linear storage; _ otherwise, the procedure will use the recurrence to compute elements of the sequence in linear number of arithmetic operations and constant storage; _ in the special case when the recurrence has constant coefficients, the procedure will compute elements of the sequence in logarithmic number of arithmetic operations and constant storage. ######################## STILL TO DO ############################ This is a list of things which should be done in future versions of Gfun. - Accept "hard" as argument instead of a list of types, this would mean "try all of them". - Use pade2 to guess. - Normalize the results (after pade2 has been installed). - Borel and invborel should probably be split into rectorec and diffeqtodiffeq with an extra argument (laplace or egf). - Use OrePoly to rewrite some of the code and add some functionnality. - Extend to q-calculus. - Extend the manipulation part to holonomic functions and sequences in several variables. - Extend to linear systems of differential equations or recurrences. ######################## FEED-BACK ############################## A mailing list gfun@inria.fr has been created. You should send email to it if you have any problem, comment, insult or if you want to be kept informed as new versions will be produced (if we find enough time). 

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