Let q=e2πiτ. Ramanujan's theta function f(a,b) is given by f(a,b)=∞∑n=−∞an(n+1)/2bn(n−1)/2 Then the following have nice q-continued fractions K4(q)=q1/8f(−q,−q3)f(−q2,−q2)=q1/8∞∏n=1(1−q4n−1)(1−q4n−3)(1−q4n−2)(1−q4n−2)K5(q)=q1/5f(−q,−q4)f(−q2,−q3)=q1/5∞∏n=1(1−q5n−1)(1−q5n−4)(1−q5n−2)(1−q5n−3)K6(q)=q1/3f(−q,−q5)f(−q3,−q3)=q1/3∞∏n=1(1−q6n−1)(1−q6n−5)(1−q6n−3)(1−q6n−3)K8(q)=q1/2f(−q,−q7)f(−q3,−q5)=q1/2∞∏n=1(1−q8n−1)(1−q8n−7)(1−q8n−3)(1−q8n−5)K12(q)=q1/1f(−q,−q11)f(−q5,−q7)=q1/1∞∏n=1(1−q12n−1)(1−q12n−11)(1−q12n−5)(1−q12n−7) 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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