I would like to know if there is a proof for the identity used in the superconformal index of 4d ${\cal N}=2$ gauge theory. In the paper by Rastelli el al, it was discovered that Eq. (10) is equal to the right hand side in the previous equation. \begin{equation} \exp\left[\sum_{n>0}\frac{1}{n}\frac{q^{n/2}}{1q^n}\chi_{[1]}(a_1^n)\chi_{[1]}(a_2^n)\chi_{[1]}(a_3^n) \right]=\frac{(q)_\infty}{1q}\prod^3_i \eta_2^{1/2}(a_i) \sum_{R}\frac{\chi_R(a_1)\chi_R(a_2)\chi_R(a_3)}{\dim_q R} \end{equation} wheret \begin{equation} \eta_2(x)=\exp\left[2\sum_{n>0} \frac{1}{n}\frac{q^n}{1q}(\chi_{[1]}^2(x)1)\right] \end{equation} Note that $\chi_R$ is a character of the irreducible representation of ${\mathfrak sl}_2$ with highest weight $R$. If there are only two characters involved, it could be seen as the Cauchy formula in representation theory. I wonder if somebody know a proof for this identity.
1) In appendix E of this paper there is an outline of a proof of this statement based on matching poles and residues on both sides of the identity. 2) One can also use the more generic arguments of a more recent paper even for the more 

