The hypergeometric functions used in this chapter are defined here. Paralleling Lebedev [50], let 
and 
be vectors of dimensions p and q respectively. Define 
. Define the single summation hypergeometrics 
by
An example of a single summation hypergeometric is 
, which has the integral representation for b>a>0 (see reference [50]
Now, given vectors 
of dimensions 
respectively, define the double summation hypergeometrics
In writing the arguments of hypergeometrics, vectors will be denoted by e.g., 
. However, when listing the components of a 1-dimensional vector the parentheses will be dropped. Further, when any of the or 
or 
subscripts are zero (which corresponds to an empty argument for that position), the empty vector argument of the hypergeometric will simply be omitted from the list of arguments.