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Green's function on lattices

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.