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Sparse Multivariate Taylor Series
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Assume we have three variables which get expressed as sparse
multivariate taylor series.::

  xts:=x::TaylorSeries Fraction Integer
  yts:=y::TaylorSeries Fraction Integer
  zts:=z::TaylorSeries Fraction Integer

These will cause traditional routines to expand in series form: ::

  t1:=sin(xts)

         1  3    1   5     1   7      1    9
     x - - x  + --- x  - ---- x  + ------ x  + O(11)
         6      120      5040      362880


We can ask for a specific coefficient, in this case, the coefficient
of the third power. ::

  coefficient(t1,3)

       1  3
     - - x
       6


And we can get that coefficient, expressed as a monomial. ::

  coefficient(t1,monomial(3,x)$IndexedExponents Symbol)

       1
     - -
       6


In a multivariate version we get a polynomial in x and y ::

  t2:=sin(xts + yts)

                  1  3   1    2   1  2    1  3
     (y + x) + (- - y  - - x y  - - x y - - x )
                  6      2        2       6
   + 
       1   5    1    4    1  2 3    1  3 2    1  4     1   5
     (--- y  + -- x y  + -- x y  + -- x y  + -- x y + --- x )
      120      24        12        12        24       120
   + 
     PAREN
              1   7    1     6    1   2 5    1   3 4    1   4 3    1   5 2
          - ---- y  - --- x y  - --- x y  - --- x y  - --- x y  - --- x y
            5040      720        240        144        144        240
        + 
             1   6      1   7
          - --- x y - ---- x
            720       5040
   + 
     PAREN
             1    9     1      8     1    2 7     1   3 6     1   4 5
          ------ y  + ----- x y  + ----- x y  + ---- x y  + ---- x y
          362880      40320        10080        4320        2880
        + 
            1   5 4     1   6 3     1    7 2     1    8       1    9
          ---- x y  + ---- x y  + ----- x y  + ----- x y + ------ x
          2880        4320        10080        40320       362880
   + 
     O(11)


We can ask for the third coefficient which is ::

  coefficient(t2,3)

       1  3   1    2   1  2    1  3
     - - y  - - x y  - - x y - - x
       6      2        2       6

And we can ask for the third coefficient of that coefficient in x ::

  coefficient(t2,monomial(3,x)$IndexedExponents Symbol)

       1
     - -
       6

And we can convert that result to a polynomial ::

  polynomial(t2,5)

      1   5    1    4     1  2   1  3     1  3   1    2     1  4   1  2
     --- y  + -- x y  + (-- x  - -)y  + (-- x  - - x)y  + (-- x  - - x  + 1)y
     120      24         12      6       12      2         24      2
   + 
      1   5   1  3
     --- x  - - x  + x
     120      6

See Also:

* )show SparseMultivariateTaylorSeries
* )display op coefficient