When I insert the following function \begin{equation} F(X,Y)=-\frac{1}{Y^{2/3}}\sum _{m=0}^{\infty } \frac{\Gamma \left(\frac{m+2}{3}\right)}{m! \Gamma (m+1)}\left(-\frac{X^2}{2^2 Y^{\frac{1}{3}}}\right)^m \end{equation} into Wolfram Mathematica, the outcome is as follows \begin{equation} F(X,Y)=-\frac{1}{192 Y^{4/3}}\left[192 Y^{2/3} \Gamma \left(\frac{2}{3}\right) \, _0F_4\left(;\frac{1}{3},\frac{1}{3},\frac{2}{3},1;-\frac{X^6}{6^6 Y}\right)+X^4 \Gamma \left(\frac{1}{3}\right) \, _0F_4\left(;1,\frac{4}{3},\frac{5}{3},\frac{5}{3};-\frac{X^6}{6^6 Y}\right)-48 X^2 Y^\frac{1}{3} \, _0F_4\left(;\frac{2}{3},\frac{2}{3},\frac{4}{3},\frac{4}{3};-\frac{X^6}{6^6 Y}\right)\right]. \end{equation} where \begin{equation} _0F_4(;b_1,b_2,b_3,b_4; -Z)=\sum_{m=0}^{\infty}\frac{\Gamma(b_1)\Gamma(b_2)\Gamma(b_3)\Gamma(b_4)}{\Gamma(b_1+m)\Gamma(b_2+m)\Gamma(b_3+m)\Gamma(b_4+m)!m}({-Z})^m. \end{equation} is the generalized hypergeometric function.
I tried to reproduce the outcome manually, but without success. How does the program do it?