Ramanujan g- and G-Functions

The Ramanujan - and -functions are defined for by the infinite products

(1)

(2)

(Ramanujan 1913-1914). Their values for positive rational numbers are called Ramanujan's class invariants or Ramanujan-Weber class invariants (Andrews and Berndt 2005, p. 66). In terms of the Ramanujan chi-function, these definitions become

			(3)
			(4)

(Andrews and Berndt 2005, p. 66), where . The functions satisfy the equalities

			(5)
			(6)
			(7)
			(8)

and can be derived using the theory of modular functions and can always be expressed as roots of algebraic equations when is rational. They are related to the Weber modular functions.

For simplicity, Ramanujan (1913-1914) tabulated for even and for odd. For , (8) allows each function to be expressed in terms of the other as

			(9)
			(10)

For , the sign before the square root in the first formula is reversed, while the second formula is unchanged. Using (5) and the above two equations allows to be computed in terms of or

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

(11)

In terms of the elliptic modulus and complementary modulus ,

			(12)
			(13)

Here,

(14)

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

(15)

For , solving for gives

			(16)
			(17)

For , the sign between the square roots in the first formula is positive, while the second formula is unchanged. Solving for and directly in terms of then gives

			(18)
			(19)

Examples of exact values include

			(20)
			(21)
			(22)
			(23)

(Ramanujan 1913-1914, p. 351). Further analytic values for small values of can be found in Ramanujan (1913-1914), Borwein and Borwein (1987), and Berndt (1998, pp. 183-322), and have been compiled by Weisstein. Ramanujan (1913-1914) contains a typographical error labeling as .

References