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Recursive definition of the Green function of a difference equation. (English. Russian original) Zbl 0742.39001
Comput. Math. Math. Phys. 31, No. 1, 120-122 (1991); translation from Zh. Vychisl. Mat. Mat. Fiz. 31, No. 1, 166-169 (1991).
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.
Reviewer: A.B.Buche (Nagpur)
##### MSC:
 39A10 Additive difference equations 60G50 Sums of independent random variables; random walks