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Thursday, October 31, 2019

Entry 39

The functions from Entry 38 obey beautiful relations of form xn+yn=1, ϑ43(q)=ϑ44(q)+ϑ42(q)a3(q)=b3(q)+c3(q)d2(q)=e2(q)+f2(q) As well as, ϑ3(q)ϑ3(q3)=ϑ4(q)ϑ4(q3)+ϑ2(q)ϑ2(q3)a(q)a(q2)=b(q)b(q2)+c(q)c(q2)d(q)d(q)=e(q)e(q)+f(q)f(q) the last of which naturally leads to the case n=2 of the three xn+yn=1 identities above. Also, ϑ3(q4)=ϑ4(q)+ϑ2(q4)a(q3)=b(q)+c(q3)d(q2)=e(q)+f(q2)
The Jacobi thetas can be expressed in terms of each other,
ϑ3(q)=ϑ4(q)+2ϑ2(q4) ϑ4(q)=2ϑ3(q4)ϑ3(q) ϑ2(q)=ϑ3(q1/4)ϑ3(q) Similarly for the Borwein thetas, a(q)=b(q)+3c(q3) 2b(q)=3a(q3)a(q) 2c(q)=a(q1/3)a(q) And also for the derived Jacobi thetas, d(q)=e(q)+4f(q2) 3e(q)=4d(q2)d(q) 3f(q)=d(q1/2)d(q)Furthermore, we have the similar, (ϑ3(q))2=1+4n=0(q4n+11q4n+1q4n+31q4n+3) a(q)=1+6n=0(q3n+11q3n+1q3n+21q3n+2) though I yet haven't found a nice analogy for d(q).

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