Some background:

The Appell functions generalise the hypergeometric function ${}_2F_1$ to two variables.

The Lauricella functions generalise this to even more variables.

The Kampé de Fériet function generalises the Appell function in a different direction. It has two variables but four arbitrary sets of parameters. It's similar to the way that ${}_pF_q$ is a generalisation of ${}_2F_1$.

It seems natural to extend this to a function that generalises Appell in both directions, with any number of variables and parameters. I'm aware of some even more general functions such as the Srivastava-Daoust function, which has lots of extra $\theta$ and $\phi$ parameters that I don't need. I'm hoping there's a name for something in between.

Basically I'm looking for a name for this function: $${}^{p+q}F^{(t)}_{r+s} \left( \begin{matrix} a_1,\cdots,a_p\colon \begin{pmatrix} b_{11} & \cdots & b_{1q}\\ \vdots && \vdots \\ b_{t1} &\cdots & b_{tq} \end{pmatrix} ; \\ c_1,\cdots,c_r\colon \begin{pmatrix} d_{11} & \cdots & d_{1s}\\ \vdots && \vdots \\ d_{t1} &\cdots & d_{ts} \end{pmatrix}; \end{matrix} x_1, \cdots, x_t \right)=\\ \sum_{n_1=0}^\infty \cdots \sum_{n_t=0}^\infty\frac{(a_1)_{\sum_j n_j}\cdots(a_p)_{\sum_j n_j}}{(c_1)_{\sum_j n_j}\cdots(c_r)_{\sum_j n_j}}\frac{\prod_{i=1}^t \prod_{j=1}^q (b_{ij})_{n_i}}{\prod_{i=1}^t \prod_{j=1}^s (d_{ij})_{n_i}}\cdot \frac{x_1^{n_1} \cdots x_t^{n_t}}{n_1! \cdots n_t!}.$$

  • $\begingroup$ Try searching a multivariable Kampé de Fériet function, but I am not sure if will be an “official” function. $\endgroup$ Dec 14 '21 at 22:42

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