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:<math>\lambda^*(x) - \lambda^*(9x) = 2[\lambda^*(x)\lambda^*(9x)]^{1/4} - 2[\lambda^*(x)\lambda^*(9x)]^{3/4}</math>
:<math>\left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{1/2} - \left[\frac{2\lambda^*(25x)}{1-\lambda^*(25x)^2}\right]^{1/2} = 2\left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{1/12}\left[\frac{2\lambda^*(25x)}{1-\lambda^*(25x)^2}\right]^{1/12} + 2\left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{5/12}\left[\frac{2\lambda^*(25x)}{1-\lambda^*(25x)^2}\right]^{5/12} </math>
:<math>[\lambda^*(x)\lambda^*(49x)]^{1/4} + [1-\lambda^*(x)^2]^{1/8}[1-\lambda^*(49x)^2]^{1/8} = 1 </math>
:<math>a^{12}-b^{12} = 4\sqrt{2}ab+22\sqrt{2}a^3b^3+44\sqrt{2}a^5b^5+44\sqrt{2}a^7b^7+22\sqrt{2}a^9b^9+4\sqrt{2}a^{11}b^{11}\, \left(a = \left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{1/12}\right) \left(b = \left[\frac{2\lambda^*(121x)}{1-\lambda^*(121x)^2}\right]^{1/12}\right) </math>
:<math>(a^2-c^2)(a^4+c^4-7a^2c^2)[(a^2-c^2)^4-a^2c^2(a^2+c^2)^2] = 8ac+8a^{13}c^{13}\, \left(a = \left[\frac{2\lambda^*(x)}{1-\lambda^*(x)^2}\right]^{1/12}\right) \left(c = \left[\frac{2\lambda^*(169x)}{1-\lambda^*(169x)^2}\right]^{1/12}\right) </math>
These are the relations between lambda-star and the Ramanujan-G-function:
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