Semi discrete problem on upper half plane

Question

I'm interested in the following problem: I know how to solve Laplace equation on the upper half plane, let's say $\mathbb{R}^2_+$ using the Poisson kernel. Assume we have somewhat of a semi-discrete problem of similar kind, where we replace the Laplacian with the semi-discrete one, i.e. $$ \Delta_h f (x_m,t) + \partial_{tt}f(x_m,t)=\frac{f(x_m+h,t)+f(x_m-h,t)-2f(x_m,t)}{h^2}+\partial_{tt}f(x_m,t), $$ where $x_m=h\cdot m$ a lattice point in $h\mathbb{Z}$. Now consider the problem $$ \begin{cases} \Delta_h f (x_m,t) + \partial_{tt}f(x_m,t)=0 \text{ in } h\mathbb{Z}\times \mathbb{R}_{>0};\\ f(x_m,0)=g(x_m) \end{cases} $$ Can we do a similar kind of "trick" as with the continuous version of this equation? I tried it with a kernel like this: $$ P(x_m,t)=\frac{1}{\pi}\frac{t}{t^2+|x_m|^2} $$ and then taking the discrete convolution $$ P(-,t)*g(x_m)=\frac{1}{\pi}\sum_{n\in \mathbb{Z};\mbox{ }n\neq m}{\frac{t\cdot g(x_n)}{t^2 + |x_m-x_n|^2}}. $$ It didn't turn out too well. Can one even do such kind of arguments in a semi-discrete setting?