Calculation of a lattice function (Q1803131)

The function of the form \[ \begin{multlined} G(a,b,c,d;m,n)=\\ {1\over 2\pi}\int_ 0^{2\pi}\int_ 0^{2\pi}{[(a+d)(b+c)+ad+4dc]^{1/2}[1- \cos(m\varphi+n\psi )]d\varphi d\psi\over a+b+c+d-a\cos\varphi- b\cos(\varphi+\psi)-c\cos(\varphi-\psi)-d\cos\psi}\end{multlined} \] 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.