FUNCTION : qseries[findpoly] - tries to find a polynomial relation between
two given q-series with degrees specified.
CALLING SEQUENCE : findpoly(x,y,q,deg1,deg2,check)
findpoly(x,y,q,deg1,deg2)
PARAMETERS : x,y - q-series
q - variable
deg1,deg2 - positive integers
check - positive integer
GLOBAL VARIABLES : X, Y
SYNOPSIS :
findpoly(x,y,q,deg1,deg2,check) returns a possible polynomial in X,Y
(with corresponding degrees deg1, deg2) which is satisfied by x,y.
check
If check is assigned then the relation is checked to O(q ).
EXAMPLES :
> with(qseries):
> read findpoly:
> x1 := radsimp(theta2(q,100)^2/theta2(q^3,40)^2):
> x2 := theta3(q,100)^2/theta3(q^3,40)^2:
> x := x1+x2:
> c := q*etaq(q,3,100)^9/etaq(q,1,100)^3:
> a := radsimp(theta3(q,100)*theta3(q^3,40)+theta2(q,100)*theta2(q^3,40)):
> c := 3*q^(1/3)*etaq(q,3,100)^3/etaq(q,1,100):
> y := radsimp(c^3/a^3):
> m := x2:
> findpoly(x,y,q,3,1,60);
WARNING: X,Y are global.
dims , 8, 18
The polynomial is
3 2
(X + 6) Y - 27 (X + 2)
Checking to order, 60
59
O(q )
3 2
(X + 6) Y - 27 (X + 2)
> findpoly(m,y,q,6,1,50);
WARNING: X,Y are global.
dims , 14, 24
The polynomial is
2 3 4
- 1/27 (X + 6 X - 3) Y + (X - 1) (X + 1)
Checking to order, 50
51
O(q )
2 3 4
- 1/27 (X + 6 X - 3) Y + (X - 1) (X + 1)
DISCUSSION :
If we define
3 3
a = theta (q) theta (q ) + theta (q) theta (q )
3 3 2 2
3
eta(3 tau)
c := 3 -----------
eta(tau)
3
c
y := ----
3
a
then it appears that
3 2
c (x + 2)
---- = 27 --------
3 3
a (x + 6)
4
(m - 1) (m + 1)
= 27 ----------------.
2 3
(m + 6 m - 3)
where
2 2
theta2(q) theta3(q)
x = ----------- + -----------
3 2 3 2
theta2(q ) theta3(q )
2
theta3(q)
and m = -----------.
3 2
theta3(q )
SEE ALSO : findhom, findnonhom, findhomcombo, findnonhomcombo