\[G_{2}(\alpha,\alpha^{\prime};\beta,\beta^{\prime};x,y)=\sum_{m,n=0}^{\infty}%
\frac{\Gamma\left(\alpha+m\right)\Gamma\left(\alpha^{\prime}+n\right)\Gamma%
\left(\beta+n-m\right)\Gamma\left(\beta^{\prime}+m-n\right)}{\Gamma\left(%
\alpha\right)\Gamma\left(\alpha^{\prime}\right)\Gamma\left(\beta\right)\Gamma%
\left(\beta^{\prime}\right)}\frac{x^{m}y^{n}}{m!n!},\]