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I'm looking for any kind of reference on a multivariable generalization of a (confluent) hypergeometric function.

To be specific, Horn's List is a list of 34 two-variable hypergeometric functions, 20 of which are confluent. Then one of these has the following series expansion:

$$\Phi_2(\beta, \beta', \gamma, x, y) = \sum_{n,m = 0}^{\infty} \frac{(\beta)_m (\beta')_n}{(\gamma)_{m+n} m! n!} x^m y^n$$

Here $(a)_n = \Gamma[a+n]/\Gamma[a]$ is the Pochhammer symbol. Now, in some odd piece of my work (physics) I somehow arrived at a series expansion that looks as:

$$\tilde{\Phi}(\beta_1, \ldots, \beta_N, \gamma, x_1, \ldots ,x_N) = \sum_{n_1,\ldots,n_N} \frac{(\beta_1)_{n_1} \cdots (\beta_N)_{n_N} }{(\gamma)_{n_1 + \cdots + n_N} n_1!\cdots n_N!} x_1^{n_1}\cdots x_N^{n_N} $$

which is like a multivariable expansion of the $\Phi_2$ function. I was wondering if anyone knows of a reference where this function is treated / defined / mentioned? Does it have a name? So far the only multivariable generalizations I found are the Lauricella hypergeometric functions, but this series isn't one of them.

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Does anyone know how to compute Lauricella functions, implemented in any package? thanks! – user88506 Jul 31 '13 at 19:42

Gradshteyn Ryzhik Academic press 1980 pg 1057

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see here. Can you provide at least a link? – draks ... Feb 21 '12 at 10:38
I have the seventh edition of this book, and the page number 1057 doesn't make sense (it's some stuff about integration). In chapter 9 they mention some multivariable hypergeometric functions, but these are either the Lauricella ones or functions of only two variables.. Do you perhaps have a newer edition? – Olaf Feb 21 '12 at 17:26
up vote 1 down vote accepted

I found a book that mentions the series I was talking about. It's the book The H-function, and in the appendix (not available through the link) they mention that the series is simply called the confluent Lauricella hypergeometric series.

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