\[A_{n}(z)=\sum_{s=0}^{n}\frac{{\left(-n\right)_{s}}{\left(n+1\right)_{s}}{\left%
(a\right)_{s}}{\left(b\right)_{s}}}{{\left(a+1\right)_{s}}{\left(b+1\right)_{s%
}}(n!)^{2}}\*{{}_{3}F_{3}}\left({-n+s,n+1+s,1\atop 1+s,a+1+s,b+1+s};-z\right),\]