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Monday, May 26, 2025

Entry 103

Level 12. Given q=e2πiτ,  the q-Pochhammer symbol (a;q)n, and Ramanujan's functions f(a,b) and f(q) discussed in Entry 92. We have the sum-product identities A(q)=f(q,q11)f(q4)=n=0q4n2+4n(q;q2)2n+1(q4;q4)2n+1=n=1(1q12n1)(1q12n11)α12B(q)=f(q5,q7)f(q4)=n=0q4n2(q;q2)2n(q4;q4)2n=n=1(1q12n5)(1q12n7)α12

where α12=(1q12n)(1q4n)

Let a=q7/8A(q) and b=q1/8B(q) then (a,b) are radicals for appropriate τ. Define their ratio a/b K12(q)=qn=1(1q12n1)(1q12n11)(1q12n5)(1q12n7) Surprisingly, this has a continued fraction (studied by Naika) based on a general form from Ramanujan for Level 4m, though its form is not as simple as the others. It is connected to the mod 6 version, K6(q)=q1/3n=1(1q6n1)(1q6n5)(1q6n3)(1q6n3)=η(τ)η3(6τ)η(2τ)η3(3τ) or the previously discussed cubic continued fraction K6(q) by the quadratic relation 

j12I=1K12(q)+K12(q)=1K6(q)K6(q2)=η3(3τ)η(4τ)η(τ)η3(12τ)=(η2(4τ)η(6τ)η(2τ)η2(12τ))2+1

 where j12I is the McKay-Thompson series of class 12I for the Monster (A187144).

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