Entry 39

The functions from Entry 38 obey beautiful relations of form $x^n+y^n=1$, $$\vartheta_3^4(q)=\vartheta_4^4(q)+\vartheta_2^4(q)$$ $$a^3(q)=b^3(q)+c^3(q)$$ $$d^2(q)=e^2(q)+f^2(q)$$ As well as, $$\vartheta_3(q)\vartheta_3(q^3)=\vartheta_4(q)\vartheta_4(q^3)+\vartheta_2(q)\vartheta_2(q^3)$$ $$a(q)a(q^2)=b(q)b(q^2)+c(q)c(q^2)$$ $$d(q)d(q)=e(q)e(q)+f(q)f(q)$$ the last of which naturally leads to the case $n=2$ of the three $x^n+y^n=1$ identities above. Also, $$\vartheta_3(q^4)=\vartheta_4(q)+\vartheta_2(q^4)$$ $$a(q^3)=b(q)+c(q^3)$$ $$d(q^2)=e(q)+f(q^2)$$

The Jacobi thetas can be expressed in terms of each other,
$$\vartheta_3(q) = \vartheta_4(q)+2\,\vartheta_2(q^4)$$ $$\vartheta_4(q) = 2\,\vartheta_3(q^4)-\vartheta_3(q)$$ $$\vartheta_2(q) = \vartheta_3(q^{1/4})-\vartheta_3(q)$$ Similarly for the Borwein thetas, $$a(q) = b(q) + 3\,c(q^3)$$ $$2\,b(q) = 3a(q^3)-a(q)$$ $$2\,c(q) = a(q^{1/3})-a(q)$$ And also for the derived Jacobi thetas, $$d(q) = e(q) + 4\,f(q^2)$$ $$3\,e(q) = 4d(q^2)-d(q)$$ $$3\,f(q) = d(q^{1/2})-d(q)$$ Furthermore, we have the similar, $$\big(\vartheta_3(q)\big)^2 = 1+4\sum_{n=0}^\infty\left(\frac{q^{4n+1}}{1-q^{4n+1}}-\frac{q^{4n+3}}{1-q^{4n+3}}\right)$$ $$\quad\quad a(q) = 1+6\sum_{n=0}^\infty\left(\frac{q^{3n+1}}{1-q^{3n+1}}-\frac{q^{3n+2}}{1-q^{3n+2}}\right)$$ though I yet haven't found a nice analogy for $d(q)$.