Consider a special function defined as:
$$f(a_1,a_2,a_3;b_1,b_2,b_3;c;x_1,x_2,x_3,y_1,y_2,y_3,z_1,z_2,z_3)=\\ =\sum_{i_1,i_2,i_3,\\j_1,j_2,j_3,=0\\k_1,k_2,k_3}^\infty \frac{(a_1)_{i_1+i_2+i_3}(a_2)_{j_1+j_2+j_3}(a_3)_{k_1+k_2+k_3}(b_1)_{i_1+j_1+k_1}(b_2)_{i_2+j_2+k_2}(b_3)_{i_3+j_3+k_3}}{i_1!i_2!i_3!j_1!j_2!j_3!k_1!k_2!k_3!(c)_{i_1+i_2+i_3+j_1+j_2+j_3+k_1+k_2+k_3}}\prod_{r=1}^3x_r^{i_r}y_r^{j_r}z_r^{k_r}$$
where $(x)_y=\Gamma(x+y)/\Gamma(x)$ is the Pochhammer symbol.
Does this function have a name, and if so what is it?
EDIT:
Note for instance a curiosity - for e.g. $a_2,a_3=0$ the function reduces to a Lauricella function:
$$f(a,0,0,b_1,b_2,b_3,c;x_1,x_2,x_3,...) = \sum_{i_1,i_2,i_3=0}^{\infty} \frac{(a)_{i_1+i_2+i_3} (b_1)_{i_1} (b_2)_{i_2} (b_3)_{i_3}} {(c)_{i_1+i_2+i_3} \,i_1! \,i_2! \,i_3!} \,x_1^{i_1}x_2^{i_2}x_3^{i_3}$$
After some search I found that this function is actually the so called hypergeometric function of type $(4,8)$. Series and integral definitions can be found around eq. (3.24) in the book "Theory of Hypergeometric Functions" by Aomoto and Kita.