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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.

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  • $\begingroup$ I found these Lauricella functions where you can have as many factorials as needed. It is just another general function though. Tell me if these work and why you want this specific G function please. $\endgroup$ Jun 11, 2021 at 2:21
  • $\begingroup$ I don't really see how the Lauricella functions express the above. However, I have found the function myself, see my answer below. I was looking for a closed form to show that it actually is just a product of regular hypergeometric functions ${}_2F_1$ for the parameters specified above. $\endgroup$
    – horropie
    Jun 12, 2021 at 21:14

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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.

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