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Thursday, October 31, 2019

Entry 38

For consistency, let the variable q be the nome's square q=e2πiτthroughout. 

I. The null Jacobi theta functions (with z=0) are,
ϑ3(q)=m=qn2=η5(2τ)η2(τ)η2(4τ)ϑ4(q)=m=(1)nqn2=η2(τ)η(2τ)ϑ2(q)=m=q(n+1/2)2=2η2(4τ)η(2τ)
II. The Borwein cubic theta functions are,
a(q)=m,n=qm2+mn+n2=η3(τ)η(3τ)+9η3(9τ)η(3τ)b(q)=m,n=ζmnqm2+mn+n2=η3(τ)η(3τ)c(q)=m,n=q(m+1/3)2+(m+1/3)(n+1/3)+(n+1/3)2=3η3(3τ)η(τ)
with a cube root of unity ζ=e2πi/3

III. The derived Jacobi theta functions are,
d(q)=ϑ44(q)+2ϑ42(q)=η8(τ)η4(2τ)+32η8(4τ)η4(2τ)e(q)=ϑ44(q)=η8(τ)η4(2τ)f(q)=12ϑ42(q1/2)=8η8(2τ)η4(τ) These obey beautiful relations discussed in Entry 39.

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