# Proving an Identity involving sums related to the $Z(N)$-Ising model

Background:
I am trying to construct meromorphic functions satisfying a number of axioms, so-called form factors which are important objects in integrable quantum models, following this paper. Building blocks for these form factors are functions $$I_{n,\mathbf{m}}(\boldsymbol{\theta})$$. I am trying to understand their derivation by hand because higher orders are too complicated to be computable by a CAS.

One example that I can compute with mathematica and try to compute by hand is: $$I_{4,21}(\theta_1, \theta_2, \theta_3, \theta_4) = \frac{1}{2} \int_{C_{\theta}} \frac{dz_1 dz_2 dz_3}{R^3} \Big( \prod_{i=1}^3 \prod_{j=1}^4 \phi(z_i - \theta_j) \Big) \frac{1}{\phi(z_{12}) \phi(z_{21})} \prod_{i=1}^2 \frac{1}{\phi(z_{i3})} ~,~~ z_{ij} := z_i - z_j$$ with $$\phi(z) = \frac{1}{\sinh\frac{z}{2} \sinh\Big(\frac{z}{2} + \frac{i\pi}{4} \Big) }$$ and $$C_\theta$$ a contour going around each $$\theta_j$$ such that $$\int_{C_\theta} \frac{dz}{R} \phi(z - \theta_j) =1$$ for $$j\in\{1,2,3,4\}$$.
So far, I managed to evaluate the above contour integration and obtained: $$I_{4,21}(\boldsymbol{\theta}) = \sum_{k=1}^4 \Big( \prod_{\substack{i=1 \\ i\neq k}}^4 \phi(\theta_{ik}) \Big) \sum_{\substack{l=1 \\ l \neq k}}^4 \Big( \prod_{\substack{i=1 \\ i\neq l,k}}^4 \phi(\theta_{li}) \Big) ~,~~ \theta_{ij} := \theta_i - \theta_j$$

My Question:
Using mathematica, I can show that (for the definition of $$\phi$$, see above): $$\sum_{k=1}^4 \Big( \prod_{\substack{i=1 \\ i\neq k}}^4 \phi(\theta_{ik}) \Big) \sum_{\substack{l=1 \\ l \neq k}}^4 \Big( \prod_{\substack{i=1 \\ i\neq l,k}}^4 \phi(\theta_{li}) \Big) =32\sqrt{2} \Big( 2 + \sum_{\substack{i,j=1\\i How can I prove this identity by Hand?

My attempt:
So far, I found the basic identity $$\phi(\theta) + \phi(-\theta) = \frac{2\sqrt{2}}{\cosh\theta}$$. The identity can be rewritten as $$I_{4,21}(\boldsymbol{\theta}) = \sum_{k=1}^4 \Big( \prod_{\substack{i=1 \\ i\neq k}}^4 \phi(\theta_{ik}) \Big) \sum_{\substack{l=1 \\ l \neq k}}^4 \Big( \prod_{\substack{i=1 \\ i\neq l,k}}^4 \phi(\theta_{li}) \Big) =\frac{1}{4} \Big( \frac{1}{\sqrt{2}} + \sum_{\substack{i,j=1\\i Also, I verified with mathematica the partial identity: $$\sum_{\substack{l=1 \\ l \neq k}}^4 \Big( \prod_{\substack{i=1 \\ i\neq l,k}}^4 \phi(\theta_{li}) \Big) = 4 \Big( 1 + \sum_{\substack{i,j=1\\ i However, I am unable to show it by hand and do not know how to proceed from here. I suspect I am missing an identity involving $$\phi$$ which makes the identity more obvious, but I do not see it.

I am thankful for any, even partial, help! I you need more detail on the question or background, please ask.