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Ramanujan g- and G-Functions

Following Ramanujan (1913-14), write

\prod_{k=1,3,5,\dots}^\infty (1+e^{-k\pi\sqrt{n}})=2^{1/4} e^{-\pi \sqrt{n}/24} G_n
\end{displaymath} (1)

\prod_{k=1,3,5,\dots}^\infty (1-e^{-k\pi\sqrt{n}})=2^{1/4} e^{-\pi \sqrt{n}/24} g_n.
\end{displaymath} (2)

These satisfy the equalities
$\displaystyle g_{4n}$ $\textstyle =$ $\displaystyle 2^{1/4}g_nG_n$ (3)
$\displaystyle G_n$ $\textstyle =$ $\displaystyle G_{1/n}$ (4)
$\displaystyle {g_n}^{-1}$ $\textstyle =$ $\displaystyle g_{4/n}$ (5)
$\displaystyle {\textstyle{1\over 4}}$ $\textstyle =$ $\displaystyle (g_nG_n)^8({G_n}^8-{g_n}^8).$ (6)

$G_n$ and $g_n$ can be derived using the theory of Modular Functions and can always be expressed as roots of algebraic equations when $n$ is Rational. For simplicity, Ramanujan tabulated $g_n$ for $n$ Even and $G_n$ for $n$ Odd. However, (6) allows $G_n$ and $g_n$ to be solved for in terms of $g_n$ and $G_n$, giving
$\displaystyle g_n$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\left({{G_n}^8+\sqrt{{G_n}^{16}-{G_n}^{-8}}\,}\right)^{1/8}$ (7)
$\displaystyle G_n$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\left({{g_n}^8+\sqrt{{g_n}^{16}+{Gg_n}^{-8}}\,}\right)^{1/8}.$ (8)

Using (3) and the above two equations allows $g_{4n}$ to be computed in terms of $g_n$ or $G_n$
2^{1/8} g_n\left({{g_n}^8+\sqrt{{g_n}^{16}+{...
\hfill {\rm for\ } n {\rm\ odd}.\cr}
\end{displaymath} (9)

In terms of the Parameter $k$ and complementary Parameter $k'$,

$\displaystyle G_n$ $\textstyle =$ $\displaystyle (2k_nk_n')^{-1/12}$ (10)
$\displaystyle g_n$ $\textstyle =$ $\displaystyle \left({k_n'^2\over 2k}\right)^{1/12}.$ (11)

\end{displaymath} (12)

is the Elliptic Lambda Function, which gives the value of $k$ for which
{K'(k)\over K(k)}=\sqrt{n}.
\end{displaymath} (13)

Solving for $\lambda^*(n)$ gives
$\displaystyle \lambda^*(n)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}[\sqrt{1+{G_n}^{-12}}-\sqrt{1-{G_n}^{-12}}\,]$ (14)
$\displaystyle \lambda^*(n)$ $\textstyle =$ $\displaystyle {g_n}^6[\sqrt{{g_n}^{12}+{g_n}^{-12}}-{g_n}^6].$ (15)

Analytic values for small values of $n$ can be found in Ramanujan (1913-1914) and Borwein and Borwein (1987), and have been compiled in Weisstein (1996). Ramanujan (1913-1914) contains a typographical error labeling $G_{465}$ as $G_{265}$.

See also G-Function


Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 139 and 298, 1987.

Ramanujan, S. ``Modular Equations and Approximations to $\pi$.'' Quart. J. Pure. Appl. Math. 45, 350-372, 1913-1914.

mathematica.gif Weisstein, E. W. ``Elliptic Singular Values.'' Mathematica notebook EllipticSingular.m.

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© 1996-9 Eric W. Weisstein