  A method to calculate exact Green's functions on lattices in various
dimensions is presented. Expressions in terms of generalized hypergeometric
functions in one or more variables are obtained for various examples by
relating the resolvent to a contour integral, evaluated using residues.
Different ways of arranging the series leads to different combinations of
hypergeometric functions providing identities involving generalized
hypergeometric functions. The method is shown to be useful for computing
Green's functions with next-nearest neighbor hopping as well.
