# Multivariable confluence hypergeometric function

1. Feb 9, 2012

### xepma

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

In particular, Horns 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 I somehow arrived at a series expansion that looks like:

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

which is like a multivariable expansion of the $\Phi_2$ function. I was wondering if anyone knows of a paper where this function is defined / mentioned? I'd like to use it as a citation.