You should start by noticing that you have

\(\displaystyle \sum_{n=-J}^{J} \mathrm e^{nx \over J} = 1 + \sum_{n=1}^J \mathrm e^{-{nx \over J}} + \mathrm e^{nx \over J} = 1 + \sum_{n=1}^J 2\cosh{nx \over J}\)

I would then multiply this by \(\displaystyle \sinh{x \over 2J}\) and use product-sum identities to simplify.

Why do so many people use such needlessly complicated notation on here? What is \(\displaystyle m_j\) all about?