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, satisfies a five-term linear recurrence relation. The asymptotic behaviour of 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 and . 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 and 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 of a general three-step random walk.

研究了完全各向异性蜂窝晶格的晶格格林函数的解析性质，其中 ( α ₁ , α ₂ , α ₃ ) 是各向异性参数 { α _j ∈ (0, ∞): j = 1, 2, 3} , w = u + i v是 ( u , v ) 平面上的复变量。这个积分定义了一个单值解析函数G ^H ( α ₁ , α ₂ , α ₃ ; w ) 假设沿实轴从u = -( α ₁ + α ₂ + α ₃ ) 到u = ( α ₁ + α ₂ + α ₃ ) 进行切割。我们证明了G ^H ( α ₁ , α ₂ , α ₃ ; w ) 是具有十个普通正则奇异点和四个表观奇异点的二阶线性微分方程的解。明显的奇点被移除通过构造一个特定的四阶微分方程。接下来，系列解决方案在哪里| w | > ( α ₁ + α ₂ + α ₃ )，并被引入。证明了，一般来说，满足一个五项线性递推关系。as n → ∞的渐近行为也成立。为了确定G ^H ( α ₁ , α ₂ , α ₃ ; w) 沿着切割的边缘，我们定义了极限函数，其中u ∈ [−( α ₁ + α ₂ + α ₃ ), ( α ₁ + α ₂ + α ₃ )]。为和建立积分表示。特别地，发现 其中J ₀ ( z )和Y ₀ ( z )分别表示第一类和第二类贝塞尔函数，并且u ∈ (0, α ₁ + α ₂ +α ₃ )。还证明了分段函数并且可以根据第一类完全椭圆积分K ( k ) 对所有u ∈ (0, α ₁ + α ₂ + α ₃ ) 进行精确的分段计算，其中k ² ≡ k ² ( α ₁ , α ₂ , α ₃ , u )是( α ₁ , α ₂ , α ₃）和你。最后，将结果应用于完全各向异性简单立方晶格的晶格格林函数，以及平面内的皮尔逊随机游动理论。特别是，评估了各种贝塞尔函数积分，以便为一般三步随机游走的平均端到端距离推导出一个新的精确公式。