Given the Jacobi theta functions ϑn(0,q) which traditionally uses the nome q=eπiτ. Define the modular lambda function α,
α=16(η(τ/2)η(2τ))8+16=(√2η(τ/2)η2(2τ)η3(τ))8=(ϑ2(0,q)ϑ3(0,q))4
Then we propose for appropriate τ such as τ=√−d that the ratios below are radicals,
(ϑ2(0,q)√2F1(12,12,1,α))4?=α(ϑ4(0,q)√2F1(12,12,1,α))4?=1−α(ϑ3(0,q)√2F1(12,12,1,α))4?=1
Adding the first two implies the third. Hence, after removing the common denominator
(ϑ2(0,q))4+(ϑ4(0,q))4=(ϑ3(0,q))4
which is known to be true. As eta quotients in the same order above,
(2η2(2τ)η(τ))4+(η2(τ2)η(τ))4=(η5(τ)η2(τ2)η2(2τ))4
Also, the equalities ϑ3(0,q)=∞∑m=−∞qm2=η5(τ)η2(τ2)η2(2τ)which has a cubic version given in Entry 156.