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

We consider the discrete Laplacian on ${\mathbb{Z}}^d$, 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.