The Appell-Series $F_1$ is given by
\begin{equation}
F_1[a;b_1, b_2; c;x,y] = \sum_{m = 0}^\infty \sum_{n = 0}^\infty \frac{ (a)_{m+n} (b_1)_{m} (b_2)_n}{(c)_{m+n}} \frac{x^m}{m!} \frac{y^n}{n!}\,,
\end{equation}
where $(a)_n$ is the usual Pochhammer-symbol or rising factorial.
I am looking for a similar function, let's call it $G$, that has an additional factor $\frac{(b_1)_m}{m!}$, i.e. something like
\begin{equation}
G[a;b_1, b_2; c;x,y] = \sum_{m = 0}^\infty \sum_{n = 0}^\infty \frac{(b_1)_m}{m!} \frac{ (a)_{m+n} (b_1)_{m} (b_2)_n}{(c)_{m+n}} \frac{x^m}{m!} \frac{y^n}{n!}\,.
\end{equation}
or of course something more general
\begin{equation}
G[a;b_1, b_2, b_3; c_1, c_2;x,y] = \sum_{m = 0}^\infty \sum_{n = 0}^\infty \frac{ (a)_{m+n} (b_1)_{m} (b_2)_n (b_3)_m}{(c_1)_{m+n} (c_2)_m} \frac{x^m}{m!} \frac{y^n}{n!}\,.
\end{equation}
The Appell-Series and also its generalization the Kampé de Fériet function lack this extra term that does not come in pairs. More specifically, I am looking for the case of $x=y$ and $a = 1$, meaning that $(a)_{n+m} = (1)_{n+m} = (n+m)!$.
I would greatly appreciate any hints on how to write the above function $G$ in terms of a known hypergeometric series.
Best Answer
The Kampé de Fériet function on Wikipedia is just a special case of \begin{align} & F^{p:q;k}_{l:m;n}\left[\begin{matrix} (a)_p : (b)_q; (c)_k \\ (\alpha)_l : (\beta)_m; (\gamma)_n \end{matrix}; x,y\right] \nonumber\\ =& \sum_{r,s = 0}^\infty \frac{\prod_{j=1}^p (a_j)_{r+s} \prod_{j=1}^q (b_j)_{r} \prod_{j=1}^k (c_j)_{s}}{\prod_{j=1}^l (\alpha_j)_{r+s} \prod_{j=1}^m (\beta_j)_{r} \prod_{j=1}^n (\gamma_j)_{s}} \frac{x^r}{r!}\frac{y^s}{s!}\,, \end{align} for $m=n$ and $q=k$. This function is given for example in (Karlsson, Wennerbeg et al.,Multiple Gaussian Hypergeometric Series. Ellis Horwood, 1985.) and is usually called Kampé de Fériet function as well.