The derivative of the fractional discrete Laplacian is an exotic Riesz potential

Let ${\mathrm{\Delta }}_N$ be the multidimensional discrete Laplacian on ${\mathbb{Z}}^N$ ($N\ge 1$). In this note, we prove that, when $N=1$, the right hand derivative of $(-{\mathrm{\Delta }}_1{)}^s$ at $0$ is an exotic discrete Riesz potential (namely, the endpoint case: the order is 0) in Stein-Wainger sense (J. Anal. Math. 2000), and when $N\ge 2$, the corresponding derivative is also an exotic discrete Riesz potential with an additional corrector. A similar conclusion for the left hand derivative case is also considered. All results obtained in this note extend the logarithmic Laplacian of Chen-Weth (Comm. PDEs. 2019) to the discrete setting.