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Tuesday, June 10, 2025

Entry 157

Given the Jacobi theta functions ϑn(0,q) which traditionally uses the nome q=eπiτ. Define the modular lambda function λ(τ),

λ(τ)=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 the three identities below and for appropriate τ such as τ=d and λ=λ(τ) that the ratios below are radicals,

(ϑ2(0,q)2F1(12,12,1,λ))4?=λ(ϑ4(0,q)2F1(12,12,1,λ))4?=1α(ϑ3(0,q)2F1(12,12,1,λ))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.

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