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

We consider the discrete Laplacian on Zd, and compute asymptotic expansions of its resolvent around thresholds embedded in continuous spectrum as well as those at end points. We prove that the resolvent has a square-root branching if d is odd, and a logarithm branching if d is even, and, moreover, obtain explicit expressions for these branching parts involving the Lauricella hypergeometric function. In order to analyze a non-degenerate threshold of general form we use an elementary step-by-step expansion procedure, less dependent on special functions.