Asymptotics of the Wigner $6j$-Symbol 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.
 A: The source for this is the book of Varshalovich et al, Quantum Theory of angular momentum.    In section 9.8 one can find the following:
$$
\left\{\begin{array}{ccc}
a&b&c\\
d+R&e+R& f+R\end{array}\right\}
\approx \frac{(-1)^{a+b+d+e}}{\sqrt{2R(2c+1)}}C^{c\gamma}_{a\alpha;b\beta}
$$
where $C^{c\gamma}_{a\alpha;b\beta}$ is a Clebsch Gordan coefficient, and where $\alpha=f-e, \beta=d-f, \gamma=d-e$.  This expression is valid in the limit where $R\gg 1$.
(I have never personally used this but with a few simple test using Mathematica gives a pretty good estimate.  For instance, with $(a,b,c,d,e,f)=(3,3,2,2,4,4)$ and $R=75$, the $6j\approx -0.01763$ while the approximate expression gives $-0.01781$.)
A: For the asymptotic behavior of the Wigner 6j-Symbol when all the coefficients but one grow, you can use the Edmonds formula. In your case it reads as:
$$
    \left\lbrace\begin{matrix} x & J_1 & J_2 \\ \frac{N}{2} & \frac{N}{2} & \frac{N}{2}\end{matrix}\right\rbrace \approx \dfrac{(-1)^{J_2+N+x}}{\sqrt{(2J_2+1)(N+1)}}d^x_{J1-J2, 0}(\phi),
$$
where $d^x_{J1-J2, 0}(\phi)$ is the small Wigner d-matrix and
$$
\cos(\phi)=\frac{1}{2}\sqrt{\dfrac{J_1(J_1+1)}{N/2(N/2+1)}}.
$$
Here it is a good reference https://aip.scitation.org/doi/10.1063/1.532474 .
