## 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:

  Bailey, W.N., Generalized hypergeometric series, (1935), Cambridge University Press Cambridge · Zbl 0011.02303  Berndt, B.C., Ramanujan’s notebooks, part II, (1989), Springer-Verlag New York · Zbl 0716.11001  Berndt, B.C., Ramanujan’s notebooks, part III, (1991), Springer-Verlag New York · Zbl 0733.11001  Berndt, B.C., Ramanujan’s notebooks, part V, (1998), Springer-Verlag New York · Zbl 0886.11001  Berndt, B.C.; Bhargava, S.; Garvan, F.G., Ramanujan’s theories of elliptic functions to alternative bases, Trans. amer. math. soc., 347, 4163-4244, (1995) · Zbl 0843.33012  Borwein, J.M.; Borwein, P.B., Pi and the AGM, (1987), Wiley New York  Borwein, J.M.; Borwein, P.B., A cubic counterpart of Jacobi’s identity and the AGM, Trans. amer. math. soc., 323, 691-701, (1991) · Zbl 0725.33014  H. H. Chan, W.-C. Liaw, and, V. Tan, Ramanujan’s class invariant λ_{n} and a new class of series for 1/π, J. London Math. Soc, to appear.  Chan, H.H.; Ong, Y.L., On Eisenstein series and ∑^{∞}m, n, −∞qm2+mn+2n2, Proc. amer. math. soc., 127, 1735-1744, (1999) · Zbl 0922.11039  Liaw, W.-C., Contributions to Ramanujan’s theories of modular equations, Ramanujan-type series for 1/π, and partitions, (1999), University of Illinois Urbana  Ramanujan, S., Modular equations and approximations to π, Quart. J. math., 45, 350-372, (1914) · JFM 45.1249.01  Ramanujan, S., Collected papers, (1927), Cambridge University Press Cambridge  Ramanujan, S., Notebooks, (1957), Tata Institute of Fundamental Research Bombay  Weber, H., Lehrbuch der algebra, (1961), Chelsea New York
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