MaplePrimes - answers and comments on Question, Generating Function from List

recurrences?

Axel Vogt — Sat, 12 Apr 2008 13:35:00 Z

Using 'listtoalgeq' fails to propose a solution and 'listtodiffeq'
has difficulties with the initial conditions.

Recursion at least gives an answer, which however is not a solution:

  #with(gfun):
 
  L:=[seq(1/(n^2+1), n=1..10)];
  
  listtorec(L,u(n)); %[1]:
  rsolve(%,{u});
  op(%): evalc(%): simplify(%);

                 2            2
   [{(2 + 2 n + n ) u(n) + (-n  - 4 n - 5) u(n + 1), u(0) = 1/2}, ogf]


                                     1
                         u(n) = ------------
                                           2
                                2 + 2 n + n


Similar problems occure for K:=map ('x -> 1/x', L) and starting the
sequence in n=0 does not allow rsolve to find a solution.

So may be at http://www.research.att.com/~njas/sequences/ you can
find hints or literature (I think there is an off line version as
well).

The last I tried was 'Guess' and 'HyperG' - these older stuff from
Bruno Gauthier (based on a MMA solution by Krattenthaler) does not
work with Maple 11, it is said to be for Maple V (I filed that once
but never used it).

recurrences?

gkokovidis — Sat, 12 Apr 2008 17:14:53 Z

It seems recurrences come close for this particular example. I will keep looking into the "gfun" package. I came across "Guess" as well but could not manage to make it work. Thank you for your effort.

Regards,
Georgios Kokovidis

Dräger Medical

solve directly

Joe Riel — Sat, 12 Apr 2008 18:28:12 Z

If you suspect that the answer is a rational polynomial, you can try to solve for the coefficients

f := 1/(k^2+3*k+1):

nn := 1:
nd := 3:

g := add(a||i*k^i,i=0..nn)/add(b||i*k^i,i=0..nd);
                                     a0 + a1 k
                        g := -------------------------
                                             2       3
                             b0 + b1 k + b2 k  + b3 k


eqs := {seq(numer(eval(g-f,k=kk)),kk=1..nn+nd+2)}:
sol := solve(eqs);
sol := {a0 = -3 b3 + b2, a1 = b3, b0 = -3 b3 + b2, b1 = -8 b3 + 3 b2, b2 = b2,

    b3 = b3}

normal(subs(sol,g));
                                      1
                                 ------------
                                  2
                                 k  + 3 k + 1

Solve directly

gkokovidis — Sat, 12 Apr 2008 18:34:50 Z

Thank you for the example. I will use it for other "test" cases and try to expand upon it.

Regards,
Georgios Kokovidis

Dräger Medical

with Guess

jakubi — Sat, 12 Apr 2008 21:56:50 Z

Tried in Maple 8:

with(GUESS):
L:=[seq(1/(n^2+1), n=1..10)]:
Guess(L,one);

    1
----------
         2
1 + _i[0]

CurveFitting

Joe Riel — Sat, 12 Apr 2008 19:05:33 Z

A nicer way to do that is with CurveFitting:

f := 1/(k^2+3*k+1):
pts := [seq([k,f],k=1..10)];
pts := [[1, 1/5], [2, 1/11], [3, 1/19], [4, 1/29], [5, 1/41], [6, 1/55],

    [7, 1/71], [8, 1/89], [9, 1/109], [10, 1/131]]

CurveFitting:-RationalInterpolation(pts,n);
                                      1
                                 ------------
                                      2
                                 1 + n  + 3 n