We relate Ramanujan's theory of elliptic functions to alternative bases to the McKay-Thompson series jn=jn(τ) for the Monster defined in Entry 145. Define,
α3(τ)=27(η(τ)η(3τ))12+27=(3(η(τ/3)η(3τ))3+3)3
Let α3=α3(τ). Then we conjecture that,
2F1(13,23,1,1−α3)2F1(13,23,1,α3)=−τ√−3 as well as
j3(τ)=27α3(1−α3)=((η(τ)η(3τ))6+33(η(3τ)η(τ))6)2=(2F1(13,23,1,α3)η2(τ)×η(τ)η(3τ))24/4
Example. Let τ=√−3. Then α3=1250(187−171⋅21/3+18⋅22/3) solves 2F1(13,23,1,1−α3)2F1(13,23,1,α3)=√3×√3
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