Proving Balanced T2 and Q2 Identities Using Modular Forms.

In our paper ''Algorithms for Finding and Proving Balanced Q2 Identities'' we pointed out that the proof algorithm in that paper had not been able to prove 1,417 of the 97,306 tentative identities which the paper's search algorithm had found. To deal with this problem, we give here a more powerful proof algorithm based on a familiar technique in the theory of modular forms. The first six sections of this paper establish the connection with modular forms. As it happens, a given balanced T2 identity in x is mapped directly to a modular form identity of weight 1 on a particular congruence subgroup. The map itself consists of simply multiplying each term of the T2 identity by the same power of x, written xĪĪ. The proof algorithm, which is then applied to this identity, consists basically of the following steps: Move the terms of the identity onto the left side of the equation and expand this side into a power series up to a predetermined degree L. If the first L ++ 1 terms of this expansion are zero, then the identity is true. When this algorithm was applied to the 1,417 recalcitrant identities, it showed they were all true, as hoped. [ABSTRACT FROM AUTHOR]