FUNCTION : findtype10 - find type 10 identities
CALLING SEQUENCE : findtype10(T)
PARAMETERS : T - positive integer
-
GLOBAL VARIABLES :
xprint,NEWJACID,RJID,SYMJID,
TT1,TT2,PROVEDFL10,EBL
SYNOPSIS :
findtype10(T) cycles through symbolic expressions
_G(a[1]) _H(b[1]) + c[1] _H(a[1]) _G(b[1])
---------------------------------------------
_G(a[2]) _HM(b[2]) + c[2] _H(a[2]) _GM(b[2])
where 2 ≤ n ≤ T, a[1]b[1]=a[2]b[2]=n, (a[1],b[1],a[2],b[2])=1,
a[1] > b[1], b[2] < a[2], c[1],c[2] in {-1,1}, and
(*)
GE(a[1]) + HE(b[1]) - (HE(a[1]) + GE(b[1])) in Z,
GE(a[2]) + HE(b[2]) - (HE(a[2]) + GE(b[2])) in Z,
and [a[1],b[1],c[1]] is not an element for the list {myramtype1
(product earlier by findtype1), using CHECKRAMIDF
to check whether the expression corresponds to a likely eta-product.
If proveit is true then provemodfuncidBATCH (from theataids
package) uses provemodfuncidBATCH to prove it.
The procedure also returns a list of [a[1],b[1],c[1],a[2],b[2],c[2]]
which give identities.
EXAMPLES :
> with(qseries):
> with(thetaids):
> with(ramarobinsids):
> xprint:=false: proveit:=true:
> G:=j->1/GetaL([1],5,j):H:=j->1/GetaL([2],5,j):
> GM:=j->1/MGetaL([1],5,j): HM:=j->1/MGetaL([2],5,j):
> GE:=j->-GetaLEXP([1],5,j):HE:=j->-GetaLEXP([2],5,j):
> findtype10(12);
*** There were NO errors. Each term was modular function on
Gamma1(60). Also -mintotord=56. To prove the identity
we need to check up to O(q^(58)).
To be on the safe side we check up to O(q^(176)).
*** The identity below is PROVED!
[6, 1, -1, 6, 1, 1]
3 2
_G(6) _H(1) - _H(6) _G(1) eta(6 tau) eta(4 tau) eta(tau)
--------------------------- = -----------------------------------
_G(6) _HM(1) + _H(6) _GM(1) 2 3
eta(12 tau) eta(3 tau) eta(2 tau)
"n=", 10
[[6, 1, -1, 6, 1, 1]]
DISCUSSION :
For G,H defined
G(6) H(1) - H(6) G(1)
-----------------------
G(6) H*(1) + H(6) G*(1)
is an eta-product and the identity is proved.
SEE ALSO :
findtype1, findtype2,
findtype3, findtype4,
findtype5, findtype6,
findtype7, findtype8,
findtype9, findtype10