----------------
| Omega Calculus |
 ----------------


This document briefly describes version 2.x of the Omega package written by Axel Riese.

The package mainly provides the functions OR and OEqR which apply the Omega>= and
Omega= operator, respectively, w.r.t. a certain variable to an expression.


The calling syntax is

                         OR[expr, z]    and    OEqR[expr, z]

or

                  OR[Omega[expr, z]]    and    OEqR[OmegaEq[expr, z]]


where z is a variable (i.e., a Mathematica symbol, an indexed symbol or a subscripted
symbol) and expr is of the form

                                       LP(z)
                        -----------------------------------
                        (1 +/- pp_1(z)) ... (1 +/- pp_d(z))

with

LP(z)         a Laurent polynomial in z,
pp_i(z)       power products (with integer exponents) in z and some other variables.



Several variables can be eliminated in one stroke by calling

           OR[expr, {z_1, ..., z_n}]    and    OEqR[expr, {z_1, ..., z_n}]

or

    OR[Omega[expr, {z_1, ..., z_n}]]    and    OEqR[OmegaEq[expr, {z_1, ..., z_n}]]


In this case expr must be of the form

                                 LP(z_1,...,z_n)
              -------------------------------------------------------
              (1 +/- pp_1(z_1,...,z_n)) ... (1 +/- pp_d(z_1,...,z_n))



Further information can be found at the Omega homepage

       -----------------------------------------------------------------------
      | http://www.risc.uni-linz.ac.at/research/combinat/risc/software/Omega/ |
       -----------------------------------------------------------------------

and in the accompanying Mathematica notebook OmegaDemo.nb.



Example:
-------

For applying Partition Analysis to (a special case of) Lecture Hall partitions
one proceeds as follows.


In[1]:= <<Omega2.m          (* load the package *)

        Axel Riese's Omega implementation version 2.27 loaded


First we transform the inequalities into the corresponding Omega generating
function expression:


In[2]:= OSum[a1^n1 a2^n2 a3^n3 a4^n4, {3 n1 >= 4 n2, 2 n2 >= 3 n3, n3 >= 2 n4}, l]

        Assuming n1 >= 0
        Assuming n2 >= 0
        Assuming n3 >= 0
        Assuming n4 >= 0

                                     1
        Omega[-----------------------------------------------, {l , l , l }]
                                     2                           1   2   3
                                a2 l                   a3 l
                        3           2        a4            3
              (1 - a1 l  ) (1 - ------) (1 - ---) (1 - -----)
                       1           4           2          3
                                 l           l          l
                                  1           3          2

And now we do the elimination:


In[3]:= OR[%]

        Eliminating l
	             1
        Eliminating l
	             3
        Eliminating l
	             2

               2        3   2     3   2        4   3        6   4
         1 + a1  a2 + a1  a2  + a1  a2  a3 + a1  a2  a3 + a1  a2  a3
Out[3]= -------------------------------------------------------------
                        4   3         4   3   2         4   3   2
        (1 - a1) (1 - a1  a2 ) (1 - a1  a2  a3 ) (1 - a1  a2  a3  a4)



=======================================================================
Axel Riese
Research Institute for Symbolic Computation
J. Kepler University Linz                     Tel: +43 (0)732 2468 9939
Altenbergerstrasse 69                         Fax: +43 (0)732 2468 9930
A-4040 Linz                      e-Mail: Axel.Riese@risc.uni-linz.ac.at
Austria                               https://risc.jku.at/m/axel-riese/
=======================================================================