Ramanujan g- and G-Functions

Following Ramanujan (1913-14), write

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

(1)

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

(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)

and

can be derived using the theory of Modular Functions and can always be expressed as roots of algebraic equations when

is Rational. For simplicity, Ramanujan tabulated

for

Even and

for

Odd. However, (6) allows

and

to be solved for in terms of

and

, 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

$\begin{displaymath} g_{4n}=\cases{ 2^{1/8} g_n\left({{g_n}^8+\sqrt{{g_n}^{16}+{... ...{-8}}\,}\right)^{1/8}\cr \hfill {\rm for\ } n {\rm\ odd}.\cr} \end{displaymath}$

(9)

In terms of the Parameter and complementary Parameter ,

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

Here,

$\begin{displaymath} k_n=\lambda^*(n) \end{displaymath}$

(12)

is the Elliptic Lambda Function, which gives the value of

for which

$\begin{displaymath} {K'(k)\over K(k)}=\sqrt{n}. \end{displaymath}$

(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

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

References

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.

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