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The cubic Chan-Chua conjecture. (English) Zbl 1211.11054

Summary: A conjecture that expresses the \(n\)th power of the cubic theta function \(a(q)=\sum_j\sum_k q^{j^2+jk+k^2}\) in terms of Eisenstein series is formulated. It is an analogue of four conjectures of H. H. Chan and K. S. Chua [Ramanujan J. 7, No. 1–3, 79–89 (2003; Zbl 1031.11022)] for powers of \(\varphi^2(q)=\sum_j\sum_k q^{j^2+k^2}\). With the help of a computer, the conjecture is shown to be true for \(6\leq n \leq 100\). It is conjectured that the result continues to hold for \(n>100\).

MSC:

11F27 Theta series; Weil representation; theta correspondences
05A19 Combinatorial identities, bijective combinatorics
11F11 Holomorphic modular forms of integral weight
33E05 Elliptic functions and integrals

Citations:

Zbl 1031.11022
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