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Actually I am looking to find the local minimum of the following function : $$F(x,y)=\frac{\Gamma(x+y+1)\Gamma(n-x-y+1)}{\Gamma(n+1)}$$

The partial derivatives of this function are:

$\begin{align} f_{xx}&=f_{yy}=f_{xy}= \dfrac{\Gamma(x+y+1) \Gamma(n-x-y+1) \left[ \psi^{0}(x+y+1)-\psi^{0}(n-x-y+1) \right] }{\Gamma(n+1)} \end{align}$

so that, regardless the stationary point:

$$D=f_{xx}f_{yy}-(f_{xy})^2=0$$

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    $\begingroup$ Where is $\psi$ from by the way? Second derivative test is basically a second order Taylor expansion. You can maybe try and go to fourth order for a possible higher order local stationarity measure. $2^4=16$ partial fourth differentials so that will probably be more tiresome to handle. $\endgroup$ – mathreadler Jun 17 '16 at 11:14
  • $\begingroup$ $\psi^{0}$ comes from the derivation of $\Gamma$ function. Can you lead me to a reference explain how to implement higher order, I tried to find but I couldn't. On the other hand, in this case $f_x=f_y$, this make finding a stationary point is trivial. $\endgroup$ – Harith Fakhrey Tahir Jun 17 '16 at 11:23

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