Notes to GeneratingFunctions version 0.65:

History:

0.1:
======================================================================
ListToList                                   (L2L)
ListToSeries                                 (L2S)
SeriesToList                                 (S2L)
SeriesToSeries                               (S2S)
RecurrenceEquationToDifferentialEquation     (RE2DE)
DifferentialEquationToRecurrenceEquation     (DE2RE)
RecurrenceEquationToList                     (RE2L)

GuessRecurrenceEquation                      (GuessRE)
GuessDifferentialEquation                    (GuessDE)
GuessRationalFunction                        (GuessRatF)
GuessAlgebraicEquation                       (GuessAE)

RecurrenceEquationPlus                       (REPlus)
DifferentialEquationPlus                     (DEPlus)
RecurrenceEquationCauchy                     (RECauchy)
DifferentialEquationCauchy                   (DECauchy)
RecurrenceEquationHadamard                   (REHadamard)
DifferentialEquationHadamard                 (DEHadamard)
AlgebraicEquationToDifferentialEquation      (AE2DE)
AlgebraicCompose                             (ACompose)
RecurrenceEquationToSubsequence              (RE2Subsequence)
RecurrenceEquationInterlace                  (REInterlace)
HomogenousRecurrenceEquation                 (HomogenousRE)
HomogenousDifferentialEquation               (HomogenousDE)

0.2:
======================================================================
DefineSequence                               (DefineS)
DefineFunction                               (DefineF)
RecurrenceEquationOut                        (REOut)
DifferentialEquationOut                      (DEOut)

Operations for sequences: +, *, PSum, Delta, Shift
Operations for functions: +, *, Integrate, D, Series

0.3:
======================================================================
RecurrenceEquationShadow                     (REShadow)

Operation for sequences: == (Equal)
Operation for functions: == (Equal)

0.4:
======================================================================
Speed up in: REPlus,DEPlus,RECauchy,DECauchy,REHadamard,DEHadamard,AE2DE,
             ACompose,REInterlace,RE2Subsequence

Initial conditions in ACompose are given in special cases.

0.5:
======================================================================
Better handling of inhomogenous DEs in RE2DE.

Changes in some procedures have been made to get output REs and DEs of
lower orders, in case that input REs or DEs are inhomogenous.

Procedure RE2Subsequence has been renamed into RESubsequence.

Handling of ACompose has been improved to work with reducible algebraic
equations.

Additional computation finds DEs and REs of least order in: REPlus, DEPlus,
    REHadamard, DECauchy, AE2DE, ACompose, REInterlace, RESubsequence.

0.6:
======================================================================
Bugs, which occured in computations with REs that have positive integer
roots in their leading polynomials, were removed. Affected procedures:
REPlus, REHadamard, REInterlace, RESubsequence.

======================================================================
0.61:
In some cases, when ACompose lost initial values, these could be saved.

======================================================================
0.62:
Minor bug in RE2L fixed.

======================================================================
0.63:
Adaptations for Mathematica 3.0 (done by Axel Riese).

Use an improved version of ENullSpace.

======================================================================
0.64:
ENullSpace update (done by Axel Riese).

======================================================================
0.65:
Made the package compatible with our summation packages (done by Axel Riese).

Incorporated latest version of ENullSpace (done by Axel Riese).

======================================================================

The pacakage contains procedures for the manipulation of holonomic functions
and sequences in one variable. It consists of four main parts.
When working with the package (at this state of the art) one should
keep the following points (shortcomings) in mind.

+ The package is not sufficiently tested.

I. Guessing Part.
-----------------
+ The input lists should contain constants or rational functions in several
  variables only.
+ GuessRatF should be done using PADE-approximations. (The available
  implementation of PADE-approx. is quite slow, however. So I did not
  use it.)

II. Implementation of Closure Properties.
------------------------------------------
+ Function for converting a cyclic RE to a holonomic RE should be added.

Although I know that I have to do a lot of testing on my own, ( and this
part of the work will surely uncover one or the other bug,) I am happy
about any feedback and criticism (about bugs, design of functions, docu-
mentation, missing functions, bad timings ...).


Christian Mallinger

01.08.1997