Lauricella hypergeometric series

Definition

Let $n \ge 1$ be an integer, let $x = (x_1, \dots, x_n) \in \C^n$, and let $a, a_i, b, b_i, c, c_i$ be complex numbers with every lower parameter $c, c_i$ outside $\set{0, -1, -2, \dots}$. Write $(x)_k$ for the Pochhammer symbol, and for a multi-index $m = (m_1, \dots, m_n) \in \N^n$ write $\abs{m} = m_1 + \cdots + m_n$, $m! = m_1! \cdots m_n!$ and $x^m = x_1^{m_1} \cdots x_n^{m_n}$. The four Lauricella series are:

1. $F_A^{(n)}(a; b_1, \dots, b_n; c_1, \dots, c_n; x) = \sum_{m} \frac{(a)_{\abs{m}} (b_1)_{m_1} \cdots (b_n)_{m_n}}{(c_1)_{m_1} \cdots (c_n)_{m_n}} \frac{x^m}{m!}$, convergent for $\abs{x_1} + \cdots + \abs{x_n} < 1$;
2. $F_B^{(n)}(a_1, \dots, a_n; b_1, \dots, b_n; c; x) = \sum_{m} \frac{(a_1)_{m_1} \cdots (a_n)_{m_n} (b_1)_{m_1} \cdots (b_n)_{m_n}}{(c)_{\abs{m}}} \frac{x^m}{m!}$, convergent for $\abs{x_i} < 1$ for all $i$;
3. $F_C^{(n)}(a; b; c_1, \dots, c_n; x) = \sum_{m} \frac{(a)_{\abs{m}} (b)_{\abs{m}}}{(c_1)_{m_1} \cdots (c_n)_{m_n}} \frac{x^m}{m!}$, convergent for $\sqrt{\abs{x_1}} + \cdots + \sqrt{\abs{x_n}} < 1$;
4. $F_D^{(n)}(a; b_1, \dots, b_n; c; x) = \sum_{m} \frac{(a)_{\abs{m}} (b_1)_{m_1} \cdots (b_n)_{m_n}}{(c)_{\abs{m}}} \frac{x^m}{m!}$, convergent for $\abs{x_i} < 1$ for all $i$.

Each sum runs over all $m \in \N^n$.