Following Ramanujan (1913-1914), write product_(k = 1, 3, 5, ...)^∞(1 + e^(-k π sqrt(n))) = 2^(1/4) e^(-π sqrt(n)/24) G_n product_(k = 1, 3, 5, ...)^∞(1 - e^(-k π sqrt(n))) = 2^(1/4) e^(-π sqrt(n)/24) g_n. These satisfy the equalities g_(4n) | = | 2^(1/4) g_n G_n G_n | = | G_(1/n) g_n^(-1) | = | g_(4/n) 1/4 | = | (g_n G_n)^8(G_n^8 - g_n^8). 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.

Barnes G-function | elliptic lambda function | Weber functions