## 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.

### MSC:

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

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