Im trying to close the following series:
\begin{equation} \sum _{m=0}^{\infty } \sum _{n=0}^{\infty } \frac{(a)_m(-a)_n}{(b)_m(b)_n}\frac{1}{m+n+c}\frac{1}{m!n!}(-z)^n z^m \end{equation}
i can write write
\begin{equation} \frac{1}{m+n+c}=\frac{\Gamma(m+n+c)}{m+n+c+1}=\frac{\Gamma(c)(c)_{m+n}}{\Gamma(c+1)(c+1)_{m+n}}=\frac{(c)_{m+n}}{c(c+1)_{m+n}} \end{equation}
this then makes
\begin{equation} \sum _{m=0}^{\infty } \sum _{n=0}^{\infty } \frac{(a)_m(-a)_n}{(b)_m(b)_n}\frac{(c)_{m+n}}{c(c+1)_{m+n}}\frac{1}{m!n!}(-z)^n z^m \end{equation}
now i tried too look at the appell function as listed in http://mathworld.wolfram.com/AppellHypergeometricFunction.html
is there some way i can write it in a form that is similar to the appell function?