A unified proof of classical Ramanujan's indentities and Jacobi's four square theorem (Q2275769)

For the complex number \(\tau\) with \(\Im(\tau)>0\), and \(q:=e^{2\pi i\tau}\), we define the Jacobi theta function \(\theta_1(z|q):=-iq^{1/8}\sum_{n\in\mathbb{Z}}(-1)^n q^{n(n+1)/2}e^{(2n+1)iz}\). In the paper under review, the author uses known identities \[ \cot^2 y-\cot^2 x+8\sum_{n=1}^\infty\frac{nq^n}{1-q^n}(\cos 2nx-\cos 2ny)=\theta^\prime_1(0|q)^2\frac{\theta_1(x-y|q)\theta_1(x+y|q)}{\theta_1(x|q)^2\theta_1(y|q)^2}, \] and \[ 2\cot x\csc^2 x-16\sum_{n=1}^\infty\frac{n^2q^n}{1-q^n}\sin 2nx=\theta^\prime_1(0|q)^3\frac{\theta_1(2x|q)}{\theta_1(x|q)^4}, \] to obtain a unified proof of some classical identities due to Ramanujan, and Jacobi's four square theorem.