So, in doing some numerical computations in QFT, I've run into the following Wigner $6j$-Symbol:
$$ \left\{ \begin{array}{ccc} x & J_1 & J_2 \\ \frac{N}{2} & \frac{N}{2} & \frac{N}{2} \\ \end{array} \right\}. $$
In the regime where $x \ll J_1,J_2,N$ and $J_1 \approx J_2 \approx N$, and $N$ is large. I would like to know if there is an asymptotic formula for such a symbol, or if one can be derived. Using symmetries we can get
$$ \left\{ \begin{array}{ccc} x & \frac{1}{2} \left(J_1+J_2\right) & \frac{1}{2} \left(J_1+J_2\right) \\ \frac{N}{2} & \frac{1}{2} \left(N+J_1-J_2\right) & \frac{1}{2} \left(N-J_1+J_2\right) \\ \end{array} \right\}. $$
Perhaps this could help, I'm really not sure.