On Ramanujan’s quartic theory of elliptic functions. (English) Zbl 1005.33009

Let \(\varphi(q):=\sum_{n=-\infty}^\infty q^{n^2}\). In the classical theory of theta-functions, a fundamental inversion formula is \[ {}_2F_1(\tfrac 12, \tfrac 12;1;x)=\varphi^2(q_2),\tag{1} \] where the relationship between \(q_2\) and \(x\) is given by \[ q_r=q_r(x)=\exp\left(-\pi \csc(\pi/r)\frac {{}_2F_1(\frac {1}{r}, \frac{r-1}r;1;1-x)} {{}_2F_1(\frac {1}{r}, \frac{r-1}r;1;x)}\right) \] with \(r=2\). Ramanujan suggested that there should be corresponding theories when \(r=3\), \(4\), or \(6\); in this paper the authors develop the theory in the case \(r=4\). In the second section, the authors develop quartic inversion formulas from which they obtain an analogue of (1) for \(r=4\). In particular, with \(\psi(q):=\sum_{n=0}^\infty q^{n(n+1)/2}\) and \[ A(q):=\varphi^4(q)+16q\psi^4(q^2), \] they show that \[ {}_2F_1(\tfrac 14, \tfrac 34;1;x)=\sqrt{A(q_4)}. \] In the third section they obtain representations of certain quartic theta functions in terms of Dedekind’s eta function. In the last section they develop methods for deriving series for \(\frac 1{\pi}\) associated with the quartic theory.


33E05 Elliptic functions and integrals
11F27 Theta series; Weil representation; theta correspondences
11F20 Dedekind eta function, Dedekind sums
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