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.

中文翻译：

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低维离散拉普拉斯算子分辨率的超几何表达式

我们提出了方格上离散拉普拉斯算子的求解的显式公式，并计算其在低维阈值附近的渐近扩展。作为副产品，我们获得了离散拉普拉斯算子基本解的封闭公式。对于证明，我们根据环绕光谱的圆盘外的C型 Appell-Lauricella 超几何函数在一般维度上表示解算器。在低维中，它简化为广义超几何函数，对于该函数，某些变换公式可用于所需的扩展。