FUNCTION : findtype1 - find type 1 identities
CALLING SEQUENCE : findtype1(T)
PARAMETERS : T - positive integer
-
GLOBAL VARIABLES :
xprint,NEWJACID,RJID,SYMJID,
TT1,TT2,PROVEDFL1,EBL
SYNOPSIS :
Before running the functions G,H,GM,HM,GE,HE must be defined.
findtype1(T) cycles through symbolic expressions
_ G(a)_H(b) + c _G(b)_H(a)
where 2 ≤ n ≤ T, ab=n, (a,b)=1, b < a, c in {-1,1}, and
(*) GE(a) + HE(b) - (GE(b) + HE(a)) in Z
using CHECKRAMIDF to check whether the expression corresponds to a
likely eta-product.
If proveit is true then provemodfuncidBATCH (from theataids
package) is used to prove it. Condition (*) eliminates the case of fractional
powers of q. The procedure also returns a list of [a,b,c] which
give identities.
NOTE: Output should be assigned myramtype1.
EXAMPLES :
> with(qseries):
> with(thetaids):
> with(ramarobinsids):
> xprint:=false: proveit:=true:
> G:=j->1/GetaL([1],10,j):H:=j->1/GetaL([3],10,j):
> GM:=j->1/MGetaL([1],10,j): HM:=j->1/MGetaL([3],10,j):
> GE:=j->-GetaLEXP([1],10,j):HE:=j->-GetaLEXP([3],10,j):
> G(1),H(1);
(13/60)
JAC(0, 10, infinity) q JAC(0, 10, infinity)
-----------------------------, -----------------------------
(23/60) JAC(3, 10, infinity)
q JAC(1, 10, infinity)
> jac2getaprod(G(1)),jac2getaprod(H(1));
1 1
---------------, ---------------
eta[10, 1](tau) eta[10, 3](tau)
> myramtype1:=findtype1(6);
*** There were NO errors. Each term was modular function on
Gamma1(60). Also -mintotord=40. To prove the identity
we need to check up to O(q^(42)).
To be on the safe side we check up to O(q^(160)).
*** The identity below is PROVED!
[6, 1, -1]
_G(6) _H(1) - _G(1) _H(6) =
3
eta(30 tau) eta(12 tau) eta(5 tau) eta(4 tau)
------------------------------------------------
2 2
eta(60 tau) eta(15 tau) eta(10 tau) eta(6 tau)
[[6, 1, -1]]
DISCUSSION :
For G,H defined G(6) H(1) - G(1) H(6) is an eta-product and
the identity is proved.
SEE ALSO :
findtype1, findtype2,
findtype3, findtype4,
findtype5, findtype6,
findtype7, findtype8,
findtype9, findtype10