what is the interpretation of $\beta_{ij}$ in the translog production function? The translog production function is defined as the following.
$$\ln y=\alpha_0+\sum_{i=1}^n\alpha_i \ln x_i+\frac{1}{2}\sum_{i=1}^n\sum_{j=1}^n\ \beta_{ij}\ln x_i\ln x_j $$

I know it is a approximation of the CES production function and simplifies to  being a logged general cobb douglas production function when every $\beta_{ij}=0$.

however in cases where every (or even a single) $\beta_{ij}\neq0$ what interpretation does it take on?

i.e. I know $\alpha_i$ is interpreted as an "output elasticity", what is $\beta_{ij}$ interpreted as?
You may interpret $\alpha_i$ as the output elasticity regarding input factor $x_i$, keeping other factors $x_j, j\neq i$ constant. $\beta_{ij}$ is an indicator for complementarities in production, i.e. how much does the addition of one percent of both factors $x_i$ and $x_j$ add to $y$. Since there is the factor 0.5, and since $\alpha_i$ and $\alpha_j$ are relevant too, the interpretation is not that straightforward.