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Calculation of a lattice function. (English. Russian original) Zbl 0777.65081
Comput. Math. Math. Phys. 32, No. 4, 553-559 (1992); translation from Zh. Vychisl. Mat. Mat. Fiz. 32, No. 4, 645-651 (1992).
The function of the form \[ \begin{split} G(a,b,c,d;m,n)=\\ {1\over 2\pi}\int_ 0^{2\pi}\int_ 0^{2\pi}{[(a+d)(b+c)+ad+4dc]^{1/2}[1- \cos(m\varphi+n\psi )]d\varphi d\psi\over a+b+c+d-a\cos\varphi- b\cos(\varphi+\psi)-c\cos(\varphi-\psi)-d\cos\psi}\end{split} \] of two integer variables \(m\), \(n\) and four real variables \(a\), \(b\), \(c\), \(d\), where \((a+d)(b+c)+ad>0\), being the potential of the random walk over a quadratic lattice with transitions to eight nearest and near-to-nearest nodes, is calculated – in some particular cases – as a solution of a system of two difference equations.
Reviewer: S.Zabek (Lublin)
65C99 Probabilistic methods, stochastic differential equations
65Q05 Numerical methods for functional equations (MSC2000)
60J45 Probabilistic potential theory
60G50 Sums of independent random variables; random walks