I'm reading the following paper
http://www2.stat.duke.edu/~scs/Courses/Stat376/Papers/GibbsFieldEst/BesagJRSSB1974.pdf
On page 7 they give the result that
$$Q(\textbf{x}) = \sum_{1 \leq i \leq n} x_iG_i(x_i) +\sum_{1 \leq i \leq j \leq n}x_ix_jG_{i,j}(x_i,x_j)+ \sum_{1 \leq i \leq j \leq k \leq n} x_ix_jx_kG_{i,j,k}(x_i,x_j,x_k)+\ldots + x_1x_2 \ldots x_nG_{1,2,\ldots,n}(x_1,x_2,\ldots,x_n).$$
The only description they give about $G$ is that $x_iG_i(x_i) = Q(0,\ldots,0,x_i,0,\ldots,0) - Q(0,0, \ldots,0)$ where they say "with analogous difference formulae for the higher order G-functions."
I'm unsure how the other higher order $G$ functions would be defined does anyone have a compact description? They also factor $x_1$ out of an expression in the next page and so it's not immediately clear to me what this returns. I would appreciate any help.