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Ramanujan g- and G-Functions


Following Ramanujan (1913-1914), write

 product_(k=1,3,5,...)^infty(1+e^(-kpisqrt(n)))=2^(1/4)e^(-pisqrt(n)/24)G_n
(1)
 product_(k=1,3,5,...)^infty(1-e^(-kpisqrt(n)))=2^(1/4)e^(-pisqrt(n)/24)g_n.
(2)

These satisfy the equalities

g_(4n)=2^(1/4)g_nG_n
(3)
G_n=G_(1/n)
(4)
g_n^(-1)=g_(4/n)
(5)
1/4=(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. They are related to the Weber functions.

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

g_n=1/2(G_n^8+sqrt(G_n^(16)-G_n^(-8)))^(1/8)
(7)
G_n=1/2(g_n^8+sqrt(g_n^(16)+g_n^(-8)))^(1/8).
(8)

Using (◇) and the above two equations allows g_(4n) to be computed in terms of g_n or G_n

 g_(4n)={2^(1/8)g_n(g_n^8+sqrt(g_n^(16)+g_n^(-8)))^(1/8);  for n even; 2^(1/8)G_n(G_n^8+sqrt(G_n^(16)-G_n^(-8)))^(1/8);  for n odd.
(9)

In terms of the parameter k and complementary parameter k^',

G_n=(2k_nk_n^')^(-1/12)
(10)
g_n=((k_n^('2))/(2k))^(1/12).
(11)

Here,

 k_n=lambda^*(n)
(12)

is the elliptic lambda function, which gives the value of k for which

 (K^'(k))/(K(k))=sqrt(n).
(13)

Solving for lambda^*(n) gives

lambda^*(n)=1/2[sqrt(1+G_n^(-12))-sqrt(1-G_n^(-12))]
(14)
lambda^*(n)=g_n^6[sqrt(g_n^(12)+g_n^(-12))-g_n^6].
(15)

Solving for G_n and g_n directly in terms of lambda^*(n) then gives

G_n=2^(-1/12)[lambda^*^2(n)-lambda^*^4(n)]^(-1/24)
(16)
g_n=2^(-1/12)[1/(lambda^*(n))-lambda^*(n)]^(1/12).
(17)

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


See also

Barnes G-Function, Elliptic Lambda Function, Weber Functions

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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.

Referenced on Wolfram|Alpha

Ramanujan g- and G-Functions

Cite this as:

Weisstein, Eric W. "Ramanujan g- and G-Functions." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Ramanujang-andG-Functions.html

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