The lattice green’s function for the square lattice

الملخص EN

A simple recurrence relation connecting the lattice Green's function G(l,m) and the first derivatives with respect to t of the lattice Green's function G(l±1,m), is presented for the square lattice.

An expressions for the lattice Green’s function of the square lattice at sites G(t,1,0), G(t,2,0) and G(t,3,0) are obtained in closed forms in the interval -2 < t < 2, which contain a sum of the complete elliptic integrals of the first and the second kind.

Differential equations for G(t,0,0) , G(t,1,0) and G(t,1,1) are obtained.