The fundamental solution \(\mathcal E\) for the difference Helmholtz equation \[ \Lambda_ \alpha {\mathcal E}(P)+ k^ 2 {\mathcal E}(P)= - \delta_ h(P),\qquad P\in \Omega \] is considered, where \(\Lambda_ \alpha v:= \partial_ 1 \overline {\partial}_ 1 v+ \partial_ 2 \overline\partial_ 2 v- \alpha h^ 2 \partial_ 1\overline\partial_ 1 \partial_ 2\overline\partial_ 2 v\) is a difference operator; \((\partial_ j v)(x):= [v(x+ he_ j)- v(x)]/h\), \((\overline\partial_ j v)(x):= [v(x)- v(x- he_ j)]/h\), \(e_ j\) is a \(j\)th base vector; \(k\) is an arbitrary complex parameter; \(\delta_ h(P)= h^{- 2}\) for \(P= (0, 0)\), \(\delta_ h(P)= 0\) for \(P\neq (0, 0)\); \(\Omega= \{P= (x, y)\mid x= mh, y= nh; m,n\in \mathbb{Z}\}\). It is assumed that \(\alpha> -{1\over 2}\). Under this condition the difference operator \(\Lambda_ \alpha\) is elliptic.
The author constructs the difference forms of the Sommerfeld radiation conditions and shows that they guarantize the existence and uniqueness of the fundamental solution \(\mathcal E\). The asymptotic expansion of \(\mathcal E\) is also analyzed.