Calculation of a lattice function. (English. Russian original) Zbl 0777.65081

Kabanovich, V. I.; Kurbatov, A. M.

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)

MSC:

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

Keywords:

lattice function; random walk; difference equations