FUNCTION : ETA[provemodfuncGAMMA0id] - prove an eta-product identity
CALLING SEQUENCE : provemodfuncGAMMA0id(etaid,N)
PARAMETERS :
etaid = sum of modular functions on Gamma[0](N)
Each term in the sum is a eta-quotient to base N.
N = positive integer
GLOBAL VARIABLES :
qcheck,modfunccheck, totcheck, _ORDS,jptmp,jpqd,eptmp,gltmp,
etaPRODL,GPL,COFS,conpres,CONTERMS,mintottmp,consL,MFLB,
xprint
SYNOPSIS :
Returns min power of q needed to verify eta-quotient identity.
Also option of proving identity by computing anough terms.
EXAMPLES :
Let
P = eta(tau)^3*eta(3*tau)^3/eta(2*tau)^3/eta(6*tau)^3
Q = eta(tau)^6*eta(6*tau)^6/eta(2*tau)^6/eta(3*tau)^6
We prove the identity
-Q + 1/Q = P + 8/P
> with(qseries):
> with(ETA);
[Ffind, Fricke, GPmake, POWERPq, POWERPqMODP, POWERq, POWERqMODP, UpLB,
cuspORD, cuspORDS, cuspORDSnotoo, cuspmake, cuspord, etaCOF, etaCONSTANT,
etaWe, etamult, etanormalid, etaprodWe, etaprodtoqseries, etaprodtoqseries2,
jacbotstar, jactopstar, mintotGAMMA0ORDS, printcuspORDS, printcuspords,
provemodfuncGAMMA0id, vetainf, vp]
> P:=eta(tau)^3*eta(3*tau)^3/eta(2*tau)^3/eta(6*tau)^3;
3 3
eta(tau) eta(3 tau)
-----------------------
3 3
eta(2 tau) eta(6 tau)
> Q:=eta(tau)^6*eta(6*tau)^6/eta(2*tau)^6/eta(3*tau)^6;
6 6
eta(tau) eta(6 tau)
-----------------------
6 6
eta(2 tau) eta(3 tau)
> EID0:=expand((-Q + 1/Q - P - 8/P));
6 6 6 6 3 3
eta(tau) eta(6 tau) eta(2 tau) eta(3 tau) eta(tau) eta(3 tau)
- ----------------------- + ----------------------- - -----------------------
6 6 6 6 3 3
eta(2 tau) eta(3 tau) eta(tau) eta(6 tau) eta(2 tau) eta(6 tau)
3 3
8 eta(2 tau) eta(6 tau)
- -------------------------
3 3
eta(tau) eta(3 tau)
> etanormalid(EID0);
We divide the eta-prod identity by 1/eta(tau)^6/eta(6*tau)^6*eta(2*tau)^6*eta(3*tau)^6.
12 12 9 3
eta(tau) eta(6 tau) eta(tau) eta(6 tau)
- ------------------------- + 1 - -----------------------
12 12 3 9
eta(2 tau) eta(3 tau) eta(3 tau) eta(2 tau)
3 9
8 eta(tau) eta(6 tau)
- -----------------------
9 3
eta(3 tau) eta(2 tau)
> EID1:=%;
> xprint:=false:
> provemodfuncGAMMA0id(EID2,6);
"TERM ", 1, "of ", 4, " *****************"
"TERM ", 2, "of ", 4, " *****************"
"TERM ", 3, "of ", 4, " *****************"
"TERM ", 4, "of ", 4, " *****************"
"mintotord = ", -2
"TO PROVE the identity we need to show that v[oo](ID) > ", 2
*** There were NO errors.
*** o Each term was modular function on
Gamma0(6).
*** o We also checked that the total order of
each term was zero.
"*** WARNING: some terms were constants. ***"
"See array CONTERMS."
To prove the identity we will need to verify if up to
q^(2).
Do you want to prove the identity? (yes/no)
You entered yes.
We verify the identity to O(q^(14)).
RESULT: The identity holds to O(q^(14)).
CONCLUSION: This proves the identity since we had only
to show that v[oo](ID) > 2.
>
DISCUSSION :
SEE ALSO :