Recursive definition of the Green function of a difference equation (Q1176924)

The authors consider the difference equation \(\sum_{k,\ell} p_{k\ell} f(k+m,\ell+n)=0\), where \(m\), \(n\) are non-zero integers, \(p_{k\ell}\) possibly complex numbers and \(f\) is a function of two variables. If \(\lambda(\varphi,\psi)=\sum_{k,\ell} p_{k,\ell}\exp(ik\varphi + i\ell \psi)\), and if \(G(m,n)=[1/(2\pi)^ 2]\int^{2\pi}_ 0\int^{2\pi}_ 0\{[\exp(\hbox{im }\varphi+\hbox{ in }\psi)]/\lambda(\varphi,\psi)\}d\varphi d\psi\) exists, then \(G(m,n)\) is the Green function. Alternatively, if \(\lambda(0,0)=0\), and \([\partial\lambda(\varphi,0)/\partial\varphi]_{\varphi=0}=0\), \([\partial \lambda(0,\psi)/\partial\psi)]_{\psi=0}=0\) and if \[ G(m,n)=[1/(2\pi)^ 2]\int^{2\pi}_ 0\int^{2\pi}_ 0\{[1- \exp(\hbox{im }\varphi+\hbox{in }\psi)]/\lambda(\varphi,\psi)\}d\varphi d\psi \] exists, then \(G(m,n)\) is the Green function. Under the above conditions, there exists a recursive solution of the type \(f(r)=\sum_{r'}a(r,r')f(r')\) where \(r\) is a vector. \(G(m,n)\) satisfy the equation \(\sum_{k,\ell}p_{k,\ell}(m\ell-nk)G(k+m,\ell+n)=0\). A 3- dimensional version of the above theory is also described. The authors indicate random walks as one of the applications of the theory.