Proving Balanced T2 and Q2 Identities Using Modular Forms

Geoffrey Apel; Richard Blecksmith; John Brillhart

doi:10.1515/integ.2011.022

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Proving Balanced T² and Q² Identities Using Modular Forms

Geoffrey Apel , Richard Blecksmith and John Brillhart

Published/Copyright: June 4, 2011

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From the journal Volume 11 Issue 3

Abstract

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 x^Ī.

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.

Keywords.: T2 and Q2 Identities; Modular Forms

Received: 2009-11-14

Accepted: 2010-01-07

Published Online: 2011-06-04

Published in Print: 2011-June

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https://doi.org/10.1515/integ.2011.022

Keywords for this article

T2 and Q2 Identities; Modular Forms