I am trying to manipulate an expression involving summations in order to take advantage of a particular identity, but my result is off by some factors of -1. How can I convert:
$f=\sum_{\kappa\psi\rho}(-1)^{2p+\psi+3r}\left(\begin{matrix}p&a&q\\\psi&\alpha&-\kappa\end{matrix}\right)\left(\begin{matrix}q&b&r\\\kappa&\beta&-\rho\end{matrix}\right)\left(\begin{matrix}r&c&p\\\rho&\gamma&-\psi\end{matrix}\right)$
into something akin to:
$f=A\sum_{\kappa\psi\rho}(-1)^{p-\psi+q-\kappa+r-\rho}\left(\begin{matrix}p&a&q\\\psi&\alpha&-\kappa\end{matrix}\right)\left(\begin{matrix}q&b&r\\\kappa&\beta&-\rho\end{matrix}\right)\left(\begin{matrix}r&c&p\\\rho&\gamma&-\psi\end{matrix}\right)?$
Where $A$ is a multiplicative factor or some expression that does not involve summations (the whole point of this exercise is to eliminate the summations, and there is an identity that simplifies the sum in the second equation to a compact closed form expression).
If I multiply by $(-1)^{-p+2\psi+q-\kappa-2r-\rho}$ then do I have to multiply the other side of the equation by $\sum_{\kappa\psi\rho}(-1)^{-p+2\psi+q-\kappa-2r-\rho}\;\;?$
Note: $\left(\begin{matrix}a&b&c\\\alpha&\beta&\gamma\end{matrix}\right)$ is a Wigner 3j symbol. As such, the following relationships between the arguments to the 3j symbol hold:
- $0 \leq a, 0 \leq b, 0 \leq c$
- Any/all permutations of the triangular inequalities $\left| a-c \right| \leq b \leq a+b$
- $\left|\alpha\right| \leq a, \left|\beta\right| \leq b, \left|\gamma\right| \leq c$
- $\alpha+\beta+\gamma=0$
Additionally, in my case all arguments to the 3j symbol are integers.