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Sunday, May 25, 2025

Entry 93

Let q=e2πiτ.  Ramanujan's theta function f(a,b) is given by f(a,b)=n=an(n+1)/2bn(n1)/2 Then the following have nice q-continued fractions K4(q)=q1/8f(q,q3)f(q2,q2)=q1/8n=1(1q4n1)(1q4n3)(1q4n2)(1q4n2)K5(q)=q1/5f(q,q4)f(q2,q3)=q1/5n=1(1q5n1)(1q5n4)(1q5n2)(1q5n3)K6(q)=q1/3f(q,q5)f(q3,q3)=q1/3n=1(1q6n1)(1q6n5)(1q6n3)(1q6n3)K8(q)=q1/2f(q,q7)f(q3,q5)=q1/2n=1(1q8n1)(1q8n7)(1q8n3)(1q8n5)K12(q)=q1/1f(q,q11)f(q5,q7)=q1/1n=1(1q12n1)(1q12n11)(1q12n5)(1q12n7) with the most famous being K5(q)=R(q) since this is the Rogers-Ramanujan continued fraction. Each will be discussed in subsequent entries.

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