On the fully anisotropic honeycomb lattice Green function, Bessel function integrals and Pearson random walks

The analytical properties of the lattice Green function for the fully anisotropic honeycomb lattice are studied, where (α ₁, α ₂, α ₃) are anisotropy parameters with {α _j ∈ (0, ∞): j = 1, 2, 3}, and w = u + iv is a complex variable in a (u, v) plane. This integral defines a single-valued analytic function G ^H(α ₁, α ₂, α ₃; w) provided that a cut is made along the real axis from u = −(α ₁ + α ₂ + α ₃) to u = (α ₁ + α ₂ + α ₃). We show that G ^H(α ₁, α ₂, α ₃; w) is a solution of a second-order linear differential equation with ten ordinary regular singular points and four apparent singular points. The apparent singularities are removed by constructing a particular differential equation of fourth order. Next, the series solution where |w| > (α ₁ + α ₂ + α ₃), and is introduced. It is proved that, in general, $r_{2n}^{\text{H}}({\alpha }_1,{\alpha }_2,{\alpha }_3)$ satisfies a five-term linear recurrence relation. The asymptotic behaviour of $r_{2n}^{\text{H}}({\alpha }_1,{\alpha }_2,{\alpha }_3)$ as n → ∞ is also established. In order to determine the behaviour of G ^H(α ₁, α ₂, α ₃; w) along the edges of the cut we define the limit function where u ∈ [−(α ₁ + α ₂ + α ₃), (α ₁ + α ₂ + α ₃)]. Integral representations are established for $G_{\text{R}}^{\text{H}}({\alpha }_1,{\alpha }_2,{\alpha }_3;u)$ and $G_{\text{I}}^{\text{H}}({\alpha }_1,{\alpha }_2,{\alpha }_3;u)$. In particular, it is found that where J ₀(z) and Y ₀(z) denote Bessel functions of the first and second kind, respectively, and u ∈ (0, α ₁ + α ₂ + α ₃). It is also demonstrated that the piecewise functions $G_{\text{R}}^{\text{H}}({\alpha }_1,{\alpha }_2,{\alpha }_3;u)$ and $G_{\text{I}}^{\text{H}}({\alpha }_1,{\alpha }_2,{\alpha }_3;u)$ can be sectionally evaluated exactly for all u ∈ (0, α ₁ + α ₂ + α ₃), in terms of complete elliptic integrals of the first kind K(k), where k ² ≡ k ²(α ₁, α ₂, α ₃, u) is a rational function of (α ₁, α ₂, α ₃) and u. Finally, applications of the results are made to the lattice Green function for the fully anisotropic simple cubic lattice, and to the theory of Pearson random walks in a plane. In particular, various Bessel function integrals are evaluated in order to derive a new exact formula for the mean end-to-end distance $\overline{r}({\alpha }_1,{\alpha }_2,{\alpha }_3)$ of a general three-step random walk. (paper)