FUNCTION :  findtype2 - find type 2 identities
                

CALLING SEQUENCE :  findtype2(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.   

        findtype2(T) cycles through symbolic expressions
        
           _G(a) _G(b) + c _H(a)_H(b)
        
        where  2 ≤ n ≤ T, ab=n, (a,b)=1,  b < a, c in {-1,1},  and
        
        (*)    GE(a) + GE(b) - (HE(a) + HE(b))  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 myramtype2.

EXAMPLES :   

>  wit(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):
>  myramtype2:=findtype2(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!
[2, 3, -1]
       _G(2) _G(3) - _H(2) _H(3) = 

                                            3           
         eta(15 tau) eta(12 tau) eta(10 tau)  eta(4 tau)
         -----------------------------------------------
                    2            2                      
         eta(30 tau)  eta(20 tau)  eta(5 tau) eta(2 tau)
                          [[2, 3, -1]]


DISCUSSION :
    For G,H defined G(2) G(3) - H(2) H(3) is an eta-product and
    the identity is proved.

SEE ALSO :  

findtype1, findtype2,
findtype3, findtype4,
findtype5, findtype6,
findtype7, findtype8,
findtype9, findtype10