Let \(f\colon \mathbb{Z}^d\to \mathbb{C}\). The \(i\)-th partial derivative of \(f\) is defined by
\[ \partial_i f(x) := \frac{1}{2}(f(x+e_i)-f(x-e_i)). \]Let \(f\colon \mathbb{Z}^d\to \mathbb{C}\). The \(i\)-th partial derivative of \(f\) is defined by
\[ \partial_i f(x) := \frac{1}{2}(f(x+e_i)-f(x-e_i)). \]