[seqfan] Re: Functional Inverse

[Simon Plouffe]

Hello,

 there is a way to resolve some cases by using solve
in Maple.

 If you have an equation like F(x) = y , if the equation
is solvable you can have directly the solution for x.

It works for simple algebraic expressions.

 I used a lot of that in my thesis when we had the first
versions of GFUN.

An example. take A001006 , motzkin numbers .

(direct output from MAPLE) please change the fonts to COURRIER

aa := [1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835,
113634, 310572, 853467, 2356779, 6536382, 18199284, 50852019,
142547559,

    400763223, 1129760415, 3192727797, 9043402501, 25669818476,
73007772802, 208023278209, 593742784829]

> listtoalgeq(%,x);
Error, (in gfun:-listtoalgeq) invalid unknown function, x
> listtoalgeq(%,f(x));

     2     2
                                                      [1 + (-1 + x)
f(x) + x  f(x) , ogf]

> op(1,%);

        2     2
                                                          1 + (-1 + x)
f(x) + x  f(x)

> %=y;

      2     2
                                                        1 + (-1 + x)
f(x) + x  f(x)  = y


> solve(%,f(x));
                                                       2      2   1/2
                         2      2   1/2
                                 1 - x + (1 - 2 x - 3 x  + 4 x  y)
  -1 + x + (1 - 2 x - 3 x  + 4 x  y)
                                 ------------------------------------,
- -------------------------------------
                                                    2
                     2
                                                 2 x
                  2 x

In this case it works well.

Is this of any help ?

best regards and bonne journée
simon plouffe