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Monday, June 9, 2025

Entry 152

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(13,23,1,1α3)2F1(13,23,1,α3)=τ3 as well as

j3(τ)=27α3(1α3)=((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(18717121/3+1822/3) solves 2F1(13,23,1,1α3)2F1(13,23,1,α3)=3×3

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