The Jacobi theta function \(\theta_1(z|\tau)\) is defined by \[ \theta_1(z|\tau)=2q^{1/8}\sum_{n=0}^\infty (-1)^nq^{n(n+1)/2}\sin(2n+1)z. \] In this very interesting article, the author proves a general identity satisfied by \(\theta_1(z|\tau)\), namely, \[ \begin{split} (f(z)-f(-z))\theta_1(x|\tau)\theta_1(y|\tau)\theta_1(x+y|\tau)\theta_1(x-y|\tau)= \\ (f(y)-f(-y))\theta_1(x|\tau)\theta_1(z|\tau)\theta_1(x+z|\tau)\theta_1(x-z|\tau) \\ - (f(x)-f(-x))\theta_1(y|\tau)\theta_1(z|\tau)\theta_1(y+z|\tau)\theta_1(y-z|\tau),\end{split} \] where \(f(z)\) is an entire function satisfying the relations \[ f(z+\pi) = -f(z) \quad\text{and}\quad f(z+\pi\tau) = -q^{-3/2}e^{-6iz}f(z). \]
At first sight, the above identity appears to be “intimidating” and “useless” but the author is able to obtain very interesting consequences from it.
He shows that \[ \theta_1^3\left(x+\frac{\pi}{3}|\tau\right)+\theta_1^3\left(x-\frac{\pi}{3}|\tau\right)- \theta_1^3\left(x|\tau\right)= 3a(q)\theta_1({3x}|{3}\tau), \] where \[ a(q) = 1+6\sum_{n=0}^\infty \left(\frac{q^{3n+1}}{1-q^{3n+1}}-\frac{q^{3n+2}}{1-q^{3n+2}}\right). \]
He also derives his original proof of the identity \[ \begin{split} 32\prod_{n=1}^\infty (1-q^n)^{10} = \sum_{ \mu,\nu\in\mathbb Z} (-1)^{\mu+\nu}q^{3\mu(\mu+1)/2+\nu(\nu+1)/6}\\ \times \left(9(2\mu+1)^3(2\nu+1)-(2\mu+1)(2\nu+1)^3\right).\end{split}\tag{1} \] For other proofs of (1), see the paper by B. C. Berndt, S. H. Chan, Z.-G. Liu and H. Yesilyurt [Q. J. Math. 55, 13–30 (2004; Zbl 1060.11063)].
Finally, he gives new proofs of identities satisfied by the Hirschhorn-Garvan-Borwein two variables cubic theta functions.
In conclusion, this is an interesting paper and I highly recommend it to those who are interested in Jacobi’s theta functions.