FUNCTION :  findtype6 - find type 6 identities
                

CALLING SEQUENCE :  findtype6(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.   

        findtype6(T) cycles through symbolic expressions
        
           _G(a)_HM(b) + c _GM(a)_H(b)
        
        where  2 ≤ n ≤ T, ab=n, (a,b)=1,  b < a, c in {-1,1},  
        and least one of a, b is even,
        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.  The procedure also returns a list 
        of [a,b,c] 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):
>  findtype6(4);
*** There were NO errors.  Each term was modular function on
    Gamma1(20). Also -mintotord=4. To prove the identity
    we need to  check up to O(q^(6)).
    To be on the safe side we check up to O(q^(44)).
*** The identity below is PROVED!
[1, 1, -1]
                                                     2    
                                        2 eta(20 tau)     
      _G(1) _HM(1) - _GM(1) _H(1) = ----------------------
                                    eta(10 tau) eta(2 tau)
*** There were NO errors.  Each term was modular function on
    Gamma1(20). Also -mintotord=4. To prove the identity
    we need to  check up to O(q^(6)).
    To be on the safe side we check up to O(q^(44)).
*** The identity below is PROVED!
[1, 1, 1]
                                                    2
                                        2 eta(4 tau) 
          _G(1) _HM(1) + _GM(1) _H(1) = -------------
                                                   2 
                                         eta(2 tau)  
WARNING: There were 2 ebasethreshold problems.
         See the global array EBL.
                    [[1, 1, -1], [1, 1, 1]]



DISCUSSION :
    For G,H defined 
    G1(1) H*(1) - G*(1) H*(1), G1(1) H*(1) + G*(1) H*(1),
    are eta-products and the identities are proved.

SEE ALSO :  

findtype6, findtype2,
findtype3, findtype6,
findtype6, findtype6,
findtype7, findtype8,
findtype9, findtype60