\[x(1-x)\frac{{\partial}^{2}{F_{3}}}{{\partial x}^{2}}+y\frac{\,{\partial}^{2}{F%
_{3}}}{\,\partial x\,\partial y}+\left(\gamma-(\alpha+\beta+1)x\right)\frac{%
\partial{F_{3}}}{\partial x}-\alpha\beta{F_{3}}=0,\]