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Search: MSC category 33C05 ( Classical hypergeometric functions, ${}_2F_1$ )

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1. CMB 1999 (vol 42 pp. 427)

Berndt, Bruce C.; Chan, Heng Huat
Ramanujan and the Modular $j$-Invariant
A new infinite product $t_n$ was introduced by S.~Ramanujan on the last page of his third notebook. In this paper, we prove Ramanujan's assertions about $t_n$ by establishing new connections between the modular $j$-invariant and Ramanujan's cubic theory of elliptic functions to alternative bases. We also show that for certain integers $n$, $t_n$ generates the Hilbert class field of $\mathbb{Q} (\sqrt{-n})$. This shows that $t_n$ is a new class invariant according to H.~Weber's definition of class invariants.

Keywords:modular functions, the Borweins' cubic theta-functions, Hilbert class fields
Categories:33C05, 33E05, 11R20, 11R29

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