# Some Connections between the Spherical and Parabolic Bases on the Cone Expressed in terms of the Macdonald Function

Authors: I. A. Shilin; Junesang Choi
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

Computing the matrix elements of the linear operator, which transforms the spherical basis ofS
O
(
3
,
1
)-representation space into the hyperbolic basis, very recently, Shilin and Choi (2013) presented an integral formula involving the product of two Legendre functions of the first kind expressed in terms of 4
F
3-hypergeometric function and, using the general Mehler-Fock transform, another integral formula for the Legendre function of the first kind. In the sequel, we investigate the pairwise connections between the spherical, hyperbolic, and parabolic bases. Using the above connections, we give an interesting series involving the Gauss hypergeometric functions expressed in terms of the Macdonald function.


---

## Body

## 1. Introduction and Preliminaries

For completeness and an easier reference, we begin by just recalling some parts of [1, Section  1]. Let Λ be the cone in the Euclidean space ℝ
4 defined by
(1)
Λ
:
=
{
(
x
1
,
x
2
,
x
3
,
x
4
)
∈
ℝ
4
∣
x
1
2
-
x
2
2
-
x
3
2
-
x
4
2
=
0
}
,
where ℝ denotes the set of real numbers. Let S
O
(
3,1
) be a multiplicative group consisting of all 4 × 4 matrices g in ℝ which satisfies the following two properties:
(2)
det
⁡
g
=
1
,
g
T
I
1,3
g
=
I
1,3
,
where g
T denotes (as usual) the transpose of the matrix g and I
1,3 is a 4 × 4 matrix given by
(3)
I
1,3
:
=
(
1
0
0
0
0
-
1
0
0
0
0
-
1
0
0
0
0
-
1
)
.Remark 1.
S
O
(
3,1
) is called the special pseudoorthogonal group. S
O
(
3,1
) is a group of linear operators preserving the quadratic form x
1
2
-
x
2
2
-
x
3
2
-
x
4
2, that is, the special classical Lorentz group. Similarly, for the group S
O
(
2,1
), see [2–4].For aσ
∈
ℂ where ℂ is the set of complex numbers, 𝔇
σ denotes the linear space consisting of infinitely differentiable σ-homogeneous functions on Λ. The representation of S
O
(
3,1
) in 𝔇
σ is a homomorphism g
↦
T
σ
(
g
), where the operator T
σ
(
g
) acts as f
(
x
)
↦
f
(
g
-
1
x
) in the space 𝔇
σ.Letγ
1 be the intersection of Λ and the hyperplane x
1
=
1, γ
2 the intersection of Λ and the pair of hyperplanes x
4
=
±
1, and γ
3 the intersection of Λ and the hyperplane x
1
+
x
4
=
0. In other words, γ
1 is a sphere with radius 2, γ
2 is a two-sheet hyperboloid, and γ
3 is a paraboloid. It is seen that, for i
∈
{
1,2
,
3
}, γ
i are two-parameter manifolds on Λ explicitly given by
(4)
γ
1
=
{
(
1
,
sin
α
1
sin
β
1
,
sin
α
1
cos
⁡
β
1
,
cos
⁡
α
1
)
∣
α
1
∈
[
0
,
π
]
,
β
1
∈
[
0
,
2
π
)
}
,
γ
2
=
{
(
cosh
⁡
α
2
,
sinh
⁡
α
2
sin
β
2
,
sinh
⁡
α
2
cos
⁡
β
2
,
±
1
)
∣
α
2
>
0
,
β
2
∈
[
0
,
2
π
)
}
,
γ
3
=
{
(
1
+
α
3
2
2
,
α
3
sin
β
3
,
α
3
cos
⁡
β
3
,
1
-
α
3
2
2
)
∣
α
3
>
0
,
β
3
∈
[
0
,
2
π
]
1
+
α
3
2
2
}
.LetH
i denote subgroups acting transitively on γ
i
(
i
=
1
,
2
,
3
). It is noted, in particular, that H
1
≃
S
O
(
3
) and H
2
≃
S
O
(
2,1
), where S
O
(
n
) is the group of rotations of n-dimensional Euclidean space ℝ
n (for more details of this group, see [5, Chapter IX]). In order to describe the group H
3, in detail, on the linear space of all diagonal matrices a
:
=
diag
⁡
(
a
1
,
a
2
), we introduce a scalar product as a
·
b
:
=
tr
⁡
(
a
b
T
). Then γ
3 is a homogeneous space of the subgroup H
3 consisting of all matrices
(5)
(
1
+
1
2
|
a
|
2
a
1
2
|
a
|
2
a
T
diag
⁡
(
1,1
)
a
T
-
1
2
|
a
|
2
-
a
1
-
1
2
|
a
|
2
)
,
where |
a
|
2 denotes the square of the length of the matrix a.TheH
1-invariant measure on γ
1 can be written as
(6)
(
d
x
)
γ
1
=
d
x
τ
(
2
)
d
x
τ
(
3
)
|
x
τ
(
4
)
|
,
where τ is an arbitrary permutation of the set {
2,3
,
4
}. The H
2-invariant measure on γ
2 is
(7)
(
d
x
)
γ
2
=
d
x
τ
(
1
)
d
x
τ
(
2
)
|
x
τ
(
3
)
|
(
τ
∈
S
3
)
,
where S
3 is the symmetric group. The H
3-invariant measure on γ
3 is
(8)
(
d
x
)
γ
3
=
d
x
2
d
x
3
.We define the bilinear functionalsF
j
:
𝔇
σ
×
𝔇
σ
^
→
ℂ
(
j
=
1
,
2
,
3
) given by
(9)
(
u
,
v
)
⟼
∫
γ
j
‍
u
(
x
)
v
(
x
)
(
d
x
)
γ
j
.Then we observe the invariant property for the functionalsF
j asserted by the following lemma.Lemma 2.
If
σ
^
=
-
σ
-
2, then, for any
i
,
j
∈
{
1,2
,
3
}, F
i
=
F
j.Proof.
Chooseτ
=
id in (6) and τ
=
(
1
2
3
) in (7) and write the measures (
d
x
)
γ
j
(
j
∈
{
1,2
}
) in (
α
j
,
β
j
)-coordinate system, respectively:
(10)
(
d
x
)
γ
1
=
|
∂
(
x
2
,
x
3
)
∂
(
α
1
,
β
1
)
|
d
α
1
d
β
1
|
x
4
|
=
sin
α
1
d
α
1
d
β
1
,
(
d
x
)
γ
2
=
|
∂
(
x
2
,
x
3
)
∂
(
α
2
,
β
2
)
|
d
α
2
d
β
2
|
x
1
|
=
sinh
⁡
α
2
d
α
2
d
β
2
.
Here id and ∂
(
x
2
,
x
3
)
/
∂
(
α
j
,
β
j
), respectively, denote the identical permutation and the corresponding Jacobian determinant.
Letu
∈
𝔇
σ and v
∈
𝔇
-
σ
-
2. Then
(11)
F
2
(
u
,
v
)
=
∫
γ
2
‍
[
sech
⁡
α
2
]
2
u
(
1
,
tanh
α
2
sin
β
2
,
tanh
α
2
cos
⁡
β
2
,
±
sech
⁡
α
2
)
·
v
(
1
,
tanh
α
2
sin
β
2
,
tanh
α
2
cos
⁡
β
2
,
±
sech
⁡
α
2
)
×
sinh
⁡
α
2
d
α
2
d
β
2
.
Setting cos
⁡
φ
:
=
±
sech
⁡
α
2, we obtain d
α
2
=
±
cosh
⁡
α
2
d
φ and sinh
⁡
α
2
=
(
1
/
2
)
sin
2
φ. We therefore find that
(12)
F
2
(
u
,
v
)
=
∫
γ
1
u
(
1
,
sin
φ
sin
β
2
,
sin
φ
cos
⁡
β
2
,
cos
⁡
φ
)
‍
.
·
v
(
1
,
sin
φ
sin
β
2
,
sin
φ
cos
⁡
β
2
,
cos
⁡
φ
)
sin
φ
d
φ
d
β
2
,
which implies F
2
=
F
1.
In the same way,(13)
F
3
(
u
,
v
)
=
4
∫
γ
3
‍
(
1
+
α
3
2
)
-
2
u
(
1
,
2
α
3
sin
β
3
1
+
α
3
2
,
2
α
3
cos
⁡
β
3
1
+
α
3
2
,
1
-
α
3
2
1
+
α
3
2
)
·
v
(
1
,
2
α
3
sin
β
3
1
+
α
3
2
,
2
α
3
cos
⁡
β
3
1
+
α
3
2
,
1
-
α
3
2
1
+
α
3
2
)
α
3
d
α
3
d
β
3
.
Setting here cos
⁡
φ
:
=
(
1
-
α
3
2
)
/
(
1
+
α
3
2
), we obtain sin
φ
=
2
α
3
2
/
(
1
+
α
3
2
) and d
α
3
=
(
(
1
+
α
3
2
)
d
φ
)
/
2. It means that
(14)
F
3
(
u
,
v
)
=
∫
γ
1
u
(
1
,
sin
φ
sin
β
3
,
sin
φ
cos
⁡
β
3
,
cos
⁡
φ
)
‍
·
v
(
1
,
sin
φ
sin
β
3
,
sin
φ
cos
⁡
β
3
,
cos
⁡
φ
)
sin
φ
d
φ
d
β
3
.
We thus see that F
1
=
F
3.
Since the relation = defined on the set{
F
1
,
F
2
,
F
3
} is transitive, we have F
2
=
F
3. The proof is complete.In [5, Chapter IX], Vilenkin constructed the canonical basis Ξ
K
l on a sphere. Here, continuing this canonical basis from the sphere γ
1 to the cone Λ via σ-homogeneity, we obtain the basis consisting of f
p
1
,
q
1:
(15)
f
p
1
,
q
1
(
x
)
=
x
1
σ
-
|
q
1
|
C
p
1
-
|
q
1
|
|
q
1
|
+
(
1
/
2
)
(
x
4
x
1
)
(
x
3
+
i
x
2
sign
⁡
q
1
)
|
q
1
|
000
(
i
=
-
1
;
p
1
,
q
1
∈
ℤ
;
p
1
≥
0
,
|
q
1
|
≤
p
1
)
,
where ℤ denotes the set of integers and C
m
p
(
t
) are the Gegenbauer polynomials (see, e.g., [6, Chapter 17]; see also [5, page 458, Equation (
3
)]). We will call it the spherical basis. Similarly, the hyperbolic basis consists of functions of the following forms:
(16)
f
p
2
,
q
2
,
±
*
(
x
)
=
(
x
4
)
±
σ
(
x
2
2
+
x
3
2
)
-
|
q
2
|
/
2
P
-
(
1
/
2
)
+
i
p
2
-
|
q
2
|
(
x
1
(
x
4
)
±
)
×
(
x
3
+
i
x
2
sign
⁡
q
2
)
|
q
2
|
(
p
2
>
0
,
q
2
∈
ℤ
)
,
where P
ν
μ
(
z
) are the Legendre functions of the first kind (see, e.g., [7, pages 194–228]; see also [8, 9]) and
(17)
(
x
)
±
μ
=
{
0
,
if
sign
⁡
x
=
∓
1
or
x
=
0
,
|
x
|
μ
,
if
sign
⁡
x
=
±
1
.
About the nonhomogeneous hyperbolic part of the hyperbolic basis, see, for instance, [5, Chapter X]. Finally, the parabolic basis consists of the functions
(18)
f
p
3
,
q
3
*
*
(
x
)
≔
(
x
1
+
x
4
)
σ
(
x
2
2
+
x
3
2
)
-
|
q
3
|
/
2
J
|
q
3
|
(
x
2
2
+
x
3
2
p
3
x
1
+
x
4
)
×
(
x
2
+
i
x
2
sign
⁡
q
3
)
|
q
3
|
,
where p
3
≥
0, q
3
∈
ℤ, and J
ν
(
z
) are the Bessel functions. The main idea of this basis is described in [5, Chapters IV and XI]; the multiplier (
x
1
+
x
4
)
σ is its σ-homogeneous part.Computing the matrix elements of the linear operator, which transforms the spherical basis ofS
O
(
3,1
)-representation space into the hyperbolic basis, very recently, Shilin and Choi [1] presented an integral formula involving the product of two Legendre functions of the first kind expressed in terms of F
4
3-hypergeometric function and, using the general Mehler-Fock transform (see, e.g., [10, 11]), another integral formula for the Legendre function of the first kind. In the sequel, using the matrix elements of the linear operators, which transform the spherical basis into the parabolic basis and the hyperbolic basis into the parabolic basis, respectively, here, in this paper, we also give certain connections between the spherical and parabolic bases on the cone and an interesting series involving the Gauss hypergeometric functions, both expressed in terms of the Macdonald function.


## 2. Description of the Connection between the Spherical and Hyperbolic Bases in terms ofF
4
3 Function

Using(19)
∫
0
2
π
‍
exp
⁡
(
i
(
p
+
q
)
z
)
d
z
=
2
π
δ
p
,
-
q
and the orthogonality relation (see, e.g., [6, page 281, Equations (
27
) and (
28
)]; see also [5, page 462, Equations (
4
) and (
5
)])
(20)
∫
-
1
1
‍
C
p
q
(
z
)
C
p
~
q
(
z
)
(
1
-
z
2
)
q
-
(
1
/
2
)
d
z
=
π
Γ
(
2
q
+
p
)
δ
p
,
p
~
2
2
q
-
1
p
!
(
p
+
q
)
[
Γ
(
q
)
]
2
,
where δ
i
,
j is the Kronecker symbol, we find from the decomposition
(21)
f
p
2
,
q
2
,
±
*
(
x
)
=
∑
p
1
=
0
∞
‍
∑
l
=
-
|
p
1
|
|
p
1
|
‍
c
(
p
2
,
q
2
,
+
)
,
(
p
1
,
l
)
f
p
1
,
l
(
x
)
that
(22)
c
(
p
2
,
q
2
,
±
)
,
(
p
1
,
l
)
=
2
2
l
-
1
(
p
1
-
l
)
!
(
p
1
+
(
1
/
2
)
)
[
Γ
(
l
+
(
1
/
2
)
)
]
2
π
2
Γ
(
2
l
+
1
)
×
F
j
(
f
p
2
,
q
2
,
+
*
,
f
p
1
,
-
l
)
,
where Γ is the familiar Gamma function (see, e.g., [12, Section 1.1]). Shilin and Choi [1] then observed that the numbers F
j
(
f
p
2
,
q
2
,
±
*
,
f
p
1
,
q
1
) give a relationship between the spherical and hyperbolic bases asserted by the following theorem.Theorem 3.
Each of the following formulas holds true:(23)
F
2
(
f
p
2
,
q
2
,
+
*
,
f
p
1
,
q
1
)
=
0
(
q
1
≠
-
q
2
)
and, for ℜ
(
σ
)
<
-
(
1
/
2
),(24)
F
2
(
f
p
2
,
-
q
1
,
+
*
,
f
p
1
,
q
1
)
=
2
p
1
-
|
q
1
|
-
σ
-
1
π
·
Γ
(
(
1
/
2
)
+
p
1
)
Γ
(
(
(
p
1
-
|
q
1
|
-
σ
+
i
p
2
)
/
2
)
-
(
1
/
4
)
)
Γ
(
(
(
p
1
-
|
q
1
|
-
σ
-
i
p
2
)
/
2
)
-
(
1
/
4
)
)
Γ
(
1
+
p
1
-
|
q
1
|
)
Γ
(
p
1
-
σ
)
Γ
(
(
1
/
2
)
+
|
q
1
|
)
·
F
4
3
[
|
q
1
|
-
p
1
2
,
1
+
|
q
1
|
-
p
1
2
,
1
+
σ
-
p
1
2
,
1
-
σ
+
p
1
2
;
1
2
-
p
1
,
5
4
+
σ
+
|
q
1
|
-
p
1
+
i
p
2
2
,
5
4
+
σ
+
|
q
1
|
-
p
1
-
i
p
2
2
;
4
]
,provided that one of the numerator parameters in F
4
3 is zero or a negative integer, and p
F
q are the generalized hypergeometric functions (see, e.g., [12, Section 1.5]).Here we provide a more general result than the one in Theorem3 for all matrix elements c
(
p
2
,
q
2
,
±
)
,
(
p
1
,
q
1
) instead of c
(
p
2
,
q
2
,
+
)
,
(
p
1
,
q
1
). We omit the corresponding proof of Theorem 3 to reformulate Theorem 3 for a general case asserted by Theorem 1
*.Theorem1
*
. Let
(25)
Ψ
ι
,
p
1
,
q
1
:
=
{
1
,
i
f
ι
=
+
o
r
p
1
-
|
q
1
|
i
s
e
v
e
n
,
-
1
,
i
f
ι
=
-
,
p
1
-
|
q
1
|
i
s
o
d
d
.
Then each of the following formulas holds true:
(26)
F
2
(
f
p
2
,
q
2
,
±
*
,
f
p
1
,
q
1
)
=
0
(
q
1
≠
-
q
2
)
and, for
ℜ
(
σ
)
<
-
(
1
/
2
)
,
(27)
F
2
(
f
p
2
,
-
q
1
,
ι
*
,
f
p
1
,
q
1
)
=
Ψ
ι
,
p
1
,
q
1
2
p
1
-
|
q
1
|
-
σ
-
1
π
·
Γ
(
(
1
/
2
)
+
p
1
)
Γ
(
(
(
p
1
-
|
q
1
|
-
σ
+
i
p
2
)
/
2
)
-
(
1
/
4
)
)
Γ
(
(
(
p
1
-
|
q
1
|
-
σ
-
i
p
2
)
/
2
)
-
(
1
/
4
)
)
Γ
(
1
+
p
1
-
|
q
1
|
)
Γ
(
p
1
-
σ
)
Γ
(
(
1
/
2
)
+
|
q
1
|
)
·
F
4
3
[
|
q
1
|
-
p
1
2
,
1
+
|
q
1
|
-
p
1
2
,
1
+
σ
-
p
1
2
,
1
-
σ
+
p
1
2
;
1
2
-
p
1
,
5
4
+
σ
+
|
q
1
|
-
p
1
+
i
p
2
2
,
5
4
+
σ
+
|
q
1
|
-
p
1
-
i
p
2
2
;
4
]
,
provided that one of the numerator parameters in
F
4
3
is zero or a negative integer, and
p
F
q
are the generalized hypergeometric functions (see, e.g.,[12, Section
1.5]).


## 3. Description of the Connection between the Spherical and Parabolic Bases in terms of the Macdonald Function

Using the same method used in Section2, from the decomposition
(28)
f
p
3
,
q
3
*
*
(
x
)
=
∑
p
1
=
0
∞
‍
∑
q
1
=
-
|
p
1
|
|
p
1
|
‍
d
(
p
3
,
q
3
)
,
(
p
1
,
q
1
)
f
p
1
,
q
1
(
x
)
,
we derive
(29)
d
(
p
3
,
q
3
)
,
(
p
1
,
q
1
)
=
2
2
|
q
1
|
-
1
(
p
1
-
|
q
1
|
)
!
(
p
1
+
(
1
/
2
)
)
[
Γ
(
|
q
1
|
+
(
1
/
2
)
)
]
2
π
2
Γ
(
2
|
q
1
|
+
1
)
×
F
j
(
f
p
3
,
q
3
*
*
,
f
p
1
,
-
|
q
1
|
)
.Theorem 4.
The following formula holds true: forℜ
(
σ
)
<
-
(
1
/
4
),
(30)
F
3
(
f
p
3
,
q
3
*
*
,
f
|
q
3
|
,
-
q
3
)
=
4
π
p
3
|
q
3
|
-
σ
-
1
[
Γ
(
|
q
3
|
-
σ
)
]
-
1
K
σ
+
1
(
p
3
)
,
where K
σ
+
1
(
p
3
) is the Macdonald function (the modified Bessel function of the second kind) (see, e.g., [13, page 675]).Proof.
We have(31)
F
3
(
f
p
3
,
q
3
*
*
,
f
p
1
,
-
q
3
)
=
2
|
q
3
|
-
σ
+
1
π
·
∫
0
+
∞
‍
α
3
|
q
3
|
+
1
(
1
+
α
3
2
)
σ
-
|
q
3
|
J
|
q
3
|
(
p
3
α
3
)
×
C
p
1
-
|
q
3
|
|
q
3
|
+
(
1
/
2
)
(
1
-
α
3
2
1
+
α
3
2
)
d
α
3
.
Using the explicit representation (see, e.g., [14, page 175, Equation (
18
)])
(32)
C
n
ν
(
x
)
=
1
Γ
(
ν
)
∑
j
=
0
[
n
/
2
]
‍
(
-
1
)
j
Γ
(
n
+
ν
-
j
)
j
!
(
n
-
2
j
)
!
(
2
x
)
n
-
2
j
,
[
u
] being the greatest integer less than or equal to u, we obtain
(33)
F
3
(
f
p
3
,
q
3
*
*
,
f
p
1
,
-
q
3
)
=
2
p
1
σ
+
1
π
[
Γ
(
|
q
3
|
+
1
2
)
]
-
1
×
∑
j
=
0
[
(
p
1
-
|
q
3
|
)
/
2
]
‍
(
-
1
)
j
2
-
2
j
Γ
(
p
1
-
j
+
(
1
/
2
)
)
j
!
(
p
1
-
|
q
3
|
-
2
j
)
!
·
∫
0
+
∞
‍
α
3
|
q
3
|
+
1
(
1
+
α
3
2
)
σ
-
p
1
+
2
j
×
(
1
-
α
3
2
)
p
1
-
|
q
3
|
-
2
j
J
|
q
3
|
(
p
3
α
3
)
d
α
3
.
In case of p
1
=
|
q
3
|, we can use the following known formula (see [15, page 179, Entry 2.12.4-28]):
(34)
∫
0
+
∞
‍
x
ν
+
1
(
x
2
+
z
2
)
ρ
J
ν
(
c
x
)
d
x
=
c
ρ
-
1
z
ν
-
ρ
+
1
2
ρ
-
1
Γ
(
ρ
)
K
ν
-
ρ
+
1
,
which holds for c
>
0, ℜ
(
z
)
>
0, and -
1
<
ℜ
(
ν
)
<
2
ℜ
(
ρ
)
-
(
1
/
2
).Corollary 5.
The following formula holds true: forℜ
(
σ
)
<
-
(
1
/
4
),
(35)
d
(
p
3
,
q
3
)
,
(
|
q
3
|
,
-
q
3
)
=
(
(
2
2
|
q
3
|
+
1
p
3
|
q
3
|
-
σ
-
1
(
|
q
3
|
+
1
2
)
[
Γ
(
|
q
3
|
+
1
2
)
]
2
)
000000
×
(
π
Γ
(
|
q
3
|
-
σ
)
Γ
(
2
|
q
3
|
+
1
)
)
-
1
[
Γ
(
|
q
3
|
+
1
2
)
]
2
)
×
K
σ
+
1
(
p
3
)
.Further we will deal only with the matrix elementsd
(
p
3
,
q
3
)
,
(
p
1
,
-
q
3
) in case of p
1
=
|
q
3
|, but let us pay attention to general case. Let ℜ
(
σ
)
<
-
(
1
/
4
). As
(36)
p
1
-
2
j
-
|
q
3
|
2
-
k
-
1
4
=
(
p
1
-
|
q
3
|
-
2
j
-
k
)
+
|
q
3
|
2
-
1
4
≥
-
1
4
,
by using the binomial theorem for the factor (
1
-
α
3
2
)
p
1
-
|
q
3
|
-
2
j in (33) and formula (34), apparently, we can represent numbers F
3
(
f
p
3
,
q
3
*
*
,
f
p
1
,
-
q
3
) as a sum whose terms contain a product of the Macdonald function and the generalized hypergeometric function. Without loss of generality, we demonstrate it, for example, under condition p
1
-
|
q
3
|
=
4. In this case,
(37)
F
3
(
f
p
3
,
q
3
*
*
,
f
p
1
,
-
q
3
)
=
4
π
p
3
p
1
-
σ
-
1
Γ
(
|
q
3
|
+
(
1
/
2
)
)
∑
j
=
0
[
(
p
1
-
|
q
3
|
)
/
2
]
‍
Ξ
j
·
Θ
j
,
where
(38)
Ξ
j
=
(
-
1
)
j
p
3
-
2
j
Γ
(
p
1
-
j
+
(
1
/
2
)
)
j
!
(
p
1
-
|
q
3
|
-
2
j
)
!
Γ
(
p
1
-
σ
-
2
j
)
,
Θ
j
=
∑
k
=
1
p
1
-
|
q
3
|
-
2
j
‍
(
-
1
)
k
(
p
1
-
|
q
3
|
-
2
j
k
)
K
σ
-
(
4
-
2
j
)
+
2
k
+
1
(
p
3
)
.
As
(39)
Γ
(
p
1
-
j
+
(
1
/
2
)
)
=
(
-
1
)
j
Γ
(
p
1
+
(
1
/
2
)
)
(
(
1
/
2
)
-
p
1
)
j
,
Γ
(
p
1
-
σ
-
2
j
)
=
2
-
2
j
Γ
(
p
1
-
σ
)
(
(
1
+
σ
-
p
1
)
/
2
)
j
(
1
-
(
(
σ
+
p
1
)
/
2
)
)
j
,
∑
k
=
0
n
‍
(
n
k
)
K
ν
±
2
k
(
z
)
=
(
-
2
)
n
K
ν
±
n
(
z
)
(regarding the last formula see, e.g., [15, Entry 4.2
.
4-2]), we have
(40)
Θ
0
=
K
σ
+
1
(
p
3
)
+
8
(
K
σ
-
1
(
p
3
)
+
K
σ
+
3
(
p
3
)
)
=
17
K
σ
+
1
(
p
3
)
+
2
σ
p
3
(
K
σ
+
2
(
p
3
)
-
K
σ
(
p
3
)
)
+
4
p
3
K
σ
+
2
(
p
3
)
=
(
17
+
4
σ
(
σ
+
1
)
p
3
2
)
K
σ
+
1
(
p
3
)
+
4
p
3
K
σ
+
2
(
p
3
)
,
Θ
1
=
2
(
σ
+
2
)
p
3
K
σ
+
2
(
p
3
)
-
2
σ
p
3
K
σ
(
p
3
)
=
4
σ
(
σ
+
1
)
p
3
2
K
σ
+
1
(
p
3
)
+
4
p
3
K
σ
+
2
(
p
3
)
,
Θ
2
=
K
σ
+
1
(
p
3
)
,
and, therefore,
(41)
F
3
(
f
p
3
,
q
3
*
*
,
f
p
1
,
-
q
3
)
=
4
π
p
3
p
1
-
σ
-
1
Γ
(
|
q
3
|
+
(
1
/
2
)
)
·
(
∑
j
=
0
∞
‍
17
Γ
(
(
1
/
2
)
+
p
1
)
Γ
(
1
+
p
1
-
|
q
3
|
)
Γ
(
p
1
-
σ
)
×
K
σ
+
1
(
p
3
)
∑
j
=
0
∞
‍
(
-
1
)
j
Ξ
j
+
4
σ
(
σ
+
1
)
Γ
(
(
1
/
2
)
+
p
1
)
p
3
2
Γ
(
1
+
p
1
-
|
q
3
|
)
Γ
(
p
1
-
σ
)
×
K
σ
+
1
(
p
3
)
∑
j
=
0
∞
‍
(
-
2
)
j
(
1
)
j
Ξ
j
+
Γ
(
p
1
-
(
3
/
2
)
)
p
3
4
Γ
(
p
1
-
|
q
3
|
-
3
)
Γ
(
p
1
-
σ
-
4
)
×
K
σ
+
1
(
p
3
)
∑
j
=
2
∞
‍
Ξ
j
+
4
Γ
(
(
1
/
2
)
+
p
1
)
p
3
Γ
(
1
+
p
1
-
|
q
3
|
)
Γ
(
p
1
-
σ
)
×
K
σ
+
2
(
p
3
)
∑
j
=
0
∞
‍
(
-
2
)
j
(
1
)
j
Ξ
j
)
.
It means that(42)
F
3
(
f
p
3
,
q
3
*
*
,
f
p
1
,
-
q
3
)
=
4
π
p
3
p
1
-
σ
-
1
Γ
(
|
q
3
|
+
1
2
)
·
(
(
|
q
3
|
-
p
1
2
,
1
+
|
q
3
|
-
p
1
2
,
1
+
σ
-
p
1
2
,
1
-
σ
+
p
1
2
,
-
2
1
2
-
p
1
,
1
|
4
p
3
2
)
17
Γ
(
(
1
/
2
)
+
p
1
)
Γ
(
1
+
p
1
-
|
q
3
|
)
Γ
(
p
1
-
σ
)
K
σ
+
1
(
p
3
)
·
F
5
1
(
|
q
3
|
-
p
1
2
,
1
+
|
q
3
|
-
p
1
2
,
1
+
σ
-
p
1
2
,
1
-
σ
+
p
1
2
,
-
1
1
2
-
p
1
|
4
p
3
2
)
+
4
σ
(
σ
+
1
)
Γ
(
(
1
/
2
)
+
p
1
)
p
3
2
Γ
(
1
+
p
1
-
|
q
3
|
)
Γ
(
p
1
-
σ
)
K
σ
+
1
(
p
3
)
·
F
5
2
(
|
q
3
|
-
p
1
2
,
1
+
|
q
3
|
-
p
1
2
,
1
+
σ
-
p
1
2
,
1
-
σ
+
p
1
2
,
-
2
1
2
-
p
1
,
1
|
4
p
3
2
)
+
Γ
(
p
1
-
(
3
/
2
)
)
p
3
4
Γ
(
p
1
-
|
q
3
|
-
3
)
Γ
(
p
1
-
σ
-
4
)
K
σ
+
1
(
p
3
)
·
F
5
2
(
2
+
|
q
3
|
-
p
1
2
,
5
+
|
q
3
|
-
p
1
2
,
5
+
σ
-
p
1
2
,
3
-
σ
+
p
1
2
,
-
1
1
2
-
p
1
,
2
|
4
p
3
2
)
+
4
Γ
(
(
1
/
2
)
+
p
1
)
p
3
Γ
(
1
+
p
1
-
|
q
3
|
)
Γ
(
p
1
-
σ
)
K
σ
+
2
(
p
3
)
·
F
5
2
(
|
q
3
|
-
p
1
2
,
1
+
|
q
3
|
-
p
1
2
,
1
+
σ
-
p
1
2
,
1
-
σ
+
p
1
2
,
-
2
1
2
-
p
1
,
1
|
4
p
3
2
)
)
.It should be remarked in passing that the same reasoning can be applied to any values ofp
1
-
|
q
3
|.


## 4. Composition of Basis Transformations and a Representation of the Macdonald Function

Letk
(
p
3
,
q
3
)
,
(
p
2
,
q
2
,
±
) be the matrix element of the linear operator acting in 𝔇
σ and mapping the hyperbolic basis into the parabolic basis; that is,
(43)
f
p
3
,
q
3
*
*
(
x
)
=
∑
q
2
∈
ℤ
‍
∫
0
+
∞
‍
k
(
p
3
,
q
3
)
,
(
p
2
,
q
2
,
±
)
f
p
2
,
q
2
,
±
*
(
x
)
d
p
2
.
The linear operator (
d
(
p
3
,
q
3
)
,
(
p
1
,
q
1
)
) can be represented as the composition of the linear operators (
c
(
p
2
,
q
2
,
±
)
,
(
p
1
,
q
1
)
) and (
k
(
p
3
,
q
3
)
,
(
p
2
,
q
2
,
±
)
); namely,
(44)
(
d
(
p
3
,
q
3
)
,
(
p
1
,
q
1
)
)
=
(
k
(
p
3
,
q
3
)
,
(
p
2
,
q
2
,
±
)
)
(
c
(
p
2
,
q
2
,
±
)
,
(
p
1
,
q
1
)
)
.
From this equality, we have, in particular,
(45)
d
(
p
3
,
q
3
)
,
(
|
q
3
|
,
-
q
3
)
=
∫
0
+
∞
‍
k
(
p
3
,
q
3
)
,
(
p
2
,
q
3
,
±
)
c
(
p
2
,
q
3
,
±
)
,
(
|
q
3
|
,
-
q
3
)
d
p
2
.Theorem 6.
The following formula holds true: forℜ
(
σ
)
<
-
(
1
/
2
),
(46)
K
σ
+
1
(
p
3
)
=
2
π
p
3
σ
Γ
(
-
σ
)
∑
n
=
0
∞
‍
(
-
1
)
n
2
2
n
+
1
[
n
!
]
2
B
(
n
+
1
,
-
2
σ
-
1
)
×
F
2
1
(
n
+
1
,
-
σ
;
n
-
2
σ
;
-
1
)
.Proof.
Note that the restrictions of the functionsf
p
2
,
q
2
,
p
m
*
∈
𝔇
-
σ
-
2 and f
p
3
,
q
3
*
*
∈
𝔇
σ to γ
3 are, respectively,
(47)
f
p
2
,
q
2
,
p
m
*
|
γ
2
≡
f
p
2
,
q
2
,
p
m
*
(
α
3
,
β
3
)
=
2
σ
+
2
(
1
-
α
3
2
)
±
P
-
(
1
/
2
)
+
i
p
2
-
|
q
2
|
(
1
+
α
3
2
1
-
α
3
2
)
exp
⁡
(
i
|
q
2
|
β
3
)
,
f
p
3
,
q
3
*
*
|
γ
2
≡
f
p
3
,
q
3
*
*
(
α
3
,
β
3
)
=
J
|
q
3
|
(
α
3
p
3
)
exp
⁡
(
i
|
q
3
|
β
3
)
.
Setting q
3
=
0, we can rewrite (45) as
(48)
K
σ
+
1
(
p
3
)
=
2
-
σ
-
1
π
p
3
σ
·
∫
0
+
∞
‍
p
2
sinh
⁡
(
π
p
2
)
Γ
(
1
2
+
i
p
2
)
Γ
(
1
2
-
i
p
2
)
×
Γ
(
-
σ
+
i
p
2
2
-
1
4
)
Γ
(
-
σ
-
i
p
2
2
-
1
4
)
·
∫
0
1
‍
α
3
(
1
-
α
3
2
)
-
σ
-
2
J
0
(
p
3
α
3
)
P
-
(
1
/
2
)
+
i
p
2
0
×
(
1
+
α
3
2
1
-
α
3
2
)
d
α
3
d
p
2
.
Changing the order of integration and using formula (see, e.g., [15, Entry 2.17.27-9])
(49)
∫
0
+
∞
‍
x
sinh
⁡
(
π
n
x
)
Γ
[
1
2
-
μ
+
i
,
1
2
-
μ
-
i
,
ν
+
i
r
x
,
ν
-
i
r
x
]
×
P
i
x
-
(
1
/
2
)
μ
(
c
)
d
x
=
R
n
,
r
(
ℜ
(
μ
)
≤
1
2
;
ℜ
(
ν
)
≥
0
)
,
R
1
,
(
1
/
2
)
=
2
(
3
/
2
)
-
2
ν
π
3
/
2
Γ
(
2
ν
-
μ
+
1
2
)
c
μ
-
2
ν
-
(
1
/
2
)
R
1
,
(
1
/
2
)
=
×
(
c
2
-
1
)
-
μ
/
2
,
we find, in the case of μ
=
0, r
=
(
1
/
2
), and ν
=
-
(
σ
/
2
)
-
(
1
/
4
), that
(50)
K
σ
+
1
(
p
3
)
=
2
π
p
3
σ
Γ
(
-
σ
)
×
∫
0
1
‍
α
3
(
1
-
α
3
2
)
-
2
σ
-
2
(
1
+
α
3
2
)
σ
J
0
(
p
3
α
3
)
d
α
3
.
Using here the decomposition
(51)
J
ν
(
z
)
=
∑
j
=
0
∞
‍
(
-
1
)
j
j
!
Γ
(
ν
+
j
+
1
)
(
z
2
)
ν
+
2
j
,
we obtain
(52)
K
σ
+
1
(
p
3
)
=
2
π
p
3
σ
Γ
(
-
σ
)
×
∑
n
=
0
∞
‍
(
-
1
)
n
2
2
n
[
n
!
]
2
∫
0
1
‍
α
3
2
n
+
1
(
1
-
α
3
2
)
-
2
σ
-
2
(
1
+
α
3
2
)
σ
d
α
3
.
To evaluate the right-sided integral, considering new variable α
3
:
=
s, we rewrite it as
(53)
1
2
∫
0
1
‍
s
n
(
1
-
s
)
-
2
σ
-
2
(
1
+
s
)
σ
d
s
and use a known formula (see, e.g., [16, page 315, Entry 3.197-8])
(54)
∫
0
a
‍
x
a
-
1
(
a
-
x
)
ω
-
1
(
x
+
z
)
-
ρ
d
x
=
a
α
+
ω
-
1
z
-
ρ
B
(
α
,
ω
)
2
F
1
(
α
,
ρ
;
α
+
ω
;
-
a
/
z
)
000
(
|
arg
z
|
<
π
;
a
>
0
;
ℜ
(
a
)
>
0
;
ℜ
(
ω
)
>
0
)
.
This completes the proof.



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*Source: 741079-2014-02-11.xml*