Exact results for the anisotropic honeycomb lattice Green function with applications to three-step Pearson random walks

The mathematical properties of the lattice Green function for the anisotropic honeycomb lattice are studied, where is a complex variable which lies in a plane, and is a real anisotropy parameter with . This double integral defines a single-valued analytic function provided that a cut is made along the real axis from to . In order to analyse the behaviour of along the edges of the cut it is convenient to define the limit function where . It is proved that the limit functions and can be sectionally evaluated exactly for all , in terms of various elliptic integrals of the first kind , where is a rational function of and u. Next, it is demonstrated that is a solution of a second order linear differential equation with eight ordinary regular singular points and two apparent singular points. It is shown that the apparent singularities can be removed by constructing a particular differential equation of third order. The series solution where and is investigated. In particular, we show that, in general, satisfies a four-term linear recurrence relation. This result is used to determine the asymptotic behaviour of as . Integral representations are established for and . It is found that where J0(z) and Y0(z) denote Bessel functions of the first and second kind, respectively, and . Finally, the results are applied to the lattice Green function for the anisotropic simple cubic lattice, and to the theory of Pearson random walks in a plane.