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We consider the discrete Laplacian on $\mathbb Z^d$, and compute asymptoticexpansions of its resolvent around thresholds embedded in continuous spectrumas well as those at end points. We prove that the resolvent has a square-rootbranching if $d$ is odd, and a logarithm branching if $d$ is even, and,moreover, obtain explicit expressions for these branching parts involving theLauricella hypergeometric function. In order to analyze a non-degeneratethreshold of general form we use an elementary step-by-step expansionprocedure, less dependent on special functions.

Branching form of the resolvent at threshold for multi-dimensional discrete Laplacians