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(1−q12n−1)(1−q12n−11)α12B(q)=f(−q5,−q7)f(−q4)=∞∑n=0q4n2(q;q2)2n(q4;q4)2n=∞∏n=1(1−q12n−5)(1−q12n−7)α12
where α12=(1−q12n)(1−q4n).
Let a=q7/8A(q) and b=q−1/8B(q) then (a,b) are radicals for appropriate τ. Define their ratio a/b K12(q)=q∞∏n=1(1−q12n−1)(1−q12n−11)(1−q12n−5)(1−q12n−7) 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)=q1/3∞∏n=1(1−q6n−1)(1−q6n−5)(1−q6n−3)(1−q6n−3)=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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