  We present an explicit formula for the resolvent of the discrete Laplacian on
the square lattice, and compute its asymptotic expansions around thresholds in
low dimensions. As a by-product we obtain a closed formula for the fundamental
solution to the discrete Laplacian. For the proofs we express the resolvent in
a general dimension in terms of the Appell--Lauricella hypergeometric function
of type $C$ outside a disk encircling the spectrum. In low dimensions it
reduces to a generalized hypergeometric function, for which certain
transformation formulas are available for the desired expansions.
