Given the nome p=eπiτ and τ=√−n, then the Ramanujan functions Gn and gn are
Gn=2−1/4p−1/24∞∏k=1,3,5,…(1+pk)gn=2−1/4p−1/24∞∏k=1,3,5,…(1−pk) These are ubiquitous in Ramanujan's Notebooks. Well-known values are G5=ϕ1/4 and G25=ϕ, where ϕ is the golden ratio. However, given the nome's square q=e2πiτ, note also the three Weber modular functions,
f(τ)=q−1/48∞∏n=1(1+qn−1/2)f1(τ)=q−1/48∞∏n=1(1−qn−1/2)f2(τ)=√2q1/24∞∏n=1(1+qn) Notice the similarity of the definitions. Mathematica doesn't have built-in functions for these, but fortunately can be expressed by the more familiar Dedekind eta function η(τ) with τ=√−n, Gn=2−1/4f(τ)=2−1/4η2(τ)η(τ2)η(2τ) gn=2−1/4f1(τ)=2−1/4η(τ2)η(τ)The two functions Gn and gn obey(Gngn)8(G8n−g8n)=14which is consequence of Weber'sf1(τ)8+f2(τ)8=f(τ)8Ramanujan calculated many explicit values for Gn and gn, one of which is the remarkable G125=3.6335…2G125φ−5/4=−1+1√5(1+22/5φ1/5(5√4−√5+(φ√5)3/2+5√4−√5−(φ√5)3/2))2 with the reciprocal golden ratio φ=−1+√52≈0.61803.
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