×

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
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bailey, W.N., Generalized hypergeometric series, (1935), Cambridge University Press Cambridge · Zbl 0011.02303
[2] Berndt, B.C., Ramanujan’s notebooks, part II, (1989), Springer-Verlag New York · Zbl 0716.11001
[3] Berndt, B.C., Ramanujan’s notebooks, part III, (1991), Springer-Verlag New York · Zbl 0733.11001
[4] Berndt, B.C., Ramanujan’s notebooks, part V, (1998), Springer-Verlag New York · Zbl 0886.11001
[5] 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
[6] Borwein, J.M.; Borwein, P.B., Pi and the AGM, (1987), Wiley New York
[7] 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
[8] 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.
[9] 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
[10] Liaw, W.-C., Contributions to Ramanujan’s theories of modular equations, Ramanujan-type series for 1/π, and partitions, (1999), University of Illinois Urbana
[11] Ramanujan, S., Modular equations and approximations to π, Quart. J. math., 45, 350-372, (1914) · JFM 45.1249.01
[12] Ramanujan, S., Collected papers, (1927), Cambridge University Press Cambridge
[13] Ramanujan, S., Notebooks, (1957), Tata Institute of Fundamental Research Bombay
[14] Weber, H., Lehrbuch der algebra, (1961), Chelsea New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.