On the fully anisotropic honeycomb lattice Green function, Bessel function integrals and Pearson random walks
Authors: Joyce G S 
Published: July 2020

The analytical properties of the lattice Green function

G

H

(

α

1

,

α

2

,

α

3

;
w

)

=

w

(

2
π

)

2

∫

−
π

π

∫

−
π

π

d

θ

1

d

θ

2

w

2

−

α

1

exp

(

i

θ

1

)

+

α

2

exp

(

i

θ

2

)

+

α

3

exp

(

−
i

θ

2

)

2

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

G

H

(

α

1

,

α

2

,

α

3

;
w

)

=

1

w

∑

n
=
0

∞

r

2
n

H

(

α

1

,

α

2

,

α

3

)

w

2
n

,

where |w| > (α
1 + α
2 + α
3), and

r

2
n

H

(

α

1

,

α

2

,

α

3

)

=

1

(

2
π

)

2

∫

−
π

π

∫

−
π

π

α

1

exp

(

i

θ

1

)

+

α

2

exp

(

i

θ

2

)

+

α

3

exp

(

−
i

θ

2

)

2
n

d

θ

1

d

θ

2

,

is introduced. It is proved that, in general,

r

2
n

H

(

α

1

,

α

2

,

α

3

)

satisfies a five-term linear recurrence relation. The asymptotic behaviour of

r

2
n

H

(

α

1

,

α

2

,

α

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

lim

ε
→
0
+

G

H

(

α

1

,

α

2

,

α

3

;
u
±
i
ε

)

≡

G

R

H

(

α

1

,

α

2

,

α

3

;
u

)

∓

i

G

I

H

(

α

1

,

α

2

,

α

3

;
u

)

,

where u ∈ [−(α
1 + α
2 + α
3), (α
1 + α
2 + α
3)]. Integral representations are established for

G

R

H

(

α

1

,

α

2

,

α

3

;
u

)

and

G

I

H

(

α

1

,

α

2

,

α

3

;
u

)

. In particular, it is found that

2

π

G

R

H

(

α

1

,

α

2

,

α

3

;
u

)

=
u

∫

0

∞

t

J

0

(

α

1

t

)

J

0

(

α

2

t

)

J

0

(

α

3

t

)

Y

0

(

u
t

)

d
t
,

2

π

G

I

H

(

α

1

,

α

2

,

α

3

;
u

)

=
u

∫

0

∞

t

J

0

(

α

1

t

)

J

0

(

α

2

t

)

J

0

(

α

3

t

)

J

0

(

u
t

)

d
t
,

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

R

H

(

α

1

,

α

2

,

α

3

;
u

)

and

G

I

H

(

α

1

,

α

2

,

α

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

r

¯

(

α

1

,

α

2

,

α

3

)

of a general three-step random walk.