Is the CES a special case of the translog production function? I was discussing today with a classmate about the relationship between prodution functions, and we tried to prove that the CES is a special case of the translog production function, but we failed. We have related the CES, the cobb-douglas and the leontief already, but we are missing the link with the translog. Is it possible to show the assumption in terms of CES parameters that yield a CES? Any help is more than welcome. 

Thanks! Jonas.
the translog is an approximation of the CES function. Let consider a 2 factors production function $A(\alpha K^\gamma+(1-\alpha)L^\gamma)^{1/\gamma}$. 
First, take the log of this production function, and then approximate this to the first-order using the McLaurin series for $\gamma$.

Using Mathematica (laziness, sorry): 

Series[Log[A(\[Alpha] K^\[Gamma]+(1-\[Alpha])L^\[Gamma])^(1/\[Gamma])], {\[Gamma], 0, 1}] // FullSimplify


You reach 
$%\log \left(A K^{\alpha } L^{1-\alpha }\right)+\frac{1}{2} \gamma(1-\alpha) \alpha  \left((\log (K)-\log (L))^2\right)+O\left(\gamma ^2\right)$

$\ln Y=\ln(A)+\alpha\ln K+(1-\alpha)\ln L+\gamma \alpha(1-\alpha)\left(\ln K+\ln L -\ln K\ln L \right)+O\left(\gamma ^2\right)$

This corresponds to the usual translog functional form used in econometric estimation.