wykresx.eps


Acta Polytechnica Vol. 51 No. 1/2011

Some Formulas for Legendre Functions Induced by the
Poisson Transform

I. A. Shilin, A. I. Nizhnikov

Abstract

Using the Poisson transform, which maps any homogeneous and infinitely differentiable function on a cone into a
corresponding function on a hyperboloid, we derive some integral representations of the Legendre functions.

Keywords: Legendre functions, Lorentz group, Poisson transform.

1 Introduction
Let us assume that the linear space Rn+1 is endowed
with the quadratic form

q(x) := x20 − x
2
1 − . . . − x

2
n.

We denote the polar bilinear form for q by q̂. The
Lorentz group SO(n,1) preserves this form and di-
vides Rn+1 into orbits. We will deal with two kinds
of these orbits. One of them is

C := {x | q(x)= 0};

it is a cone. The second kind of orbits consist of
two-sheet hyperboloids

H(r) := {x | q(x)= r2}

for any r > 0.
The group SO(n,1) has 2 connected components.

One of them contains the identity and will be under
our consideration further. We denote this subgroup
by symbol G. The action x �−→ g−1x of the group
G is transitive on C. Let σ ∈ C and Dσ be a lin-
ear subspace in C∞(C) consisting of σ-homogeneous
functions. It is useful to suppose throughout this
paper that −n +1 < re σ < 0. We define the repre-
sentation Tσ in Dσ by left shifts:

Tσ(g)[f(x)] := f(g
−1x).

Suppose that γ is a contour on C intersecting all
generatrices (i.e. all lines containing the origin). Ev-
erypoint x ∈ γ depends on n−1parameters, so every
point x ∈ C can be represented as

xi = {tFi(ξ1, . . . , ξn−1), i =1, . . . , n +1.

Denoting by G̃ the subgroup of G which acts transi-
tively on γ, we have

dx = tn−3dtdγ, (1)

where dγ is the G̃-invariantmeasure on γ.
For anypair (Dσ, Dσ̃), wedefine thebilinear func-

tionals Fγ : (Dσ, Dσ̃) −→ C,

(f1, f2) �−→
∫

γ

f1(x)f2(x)dγ.

The functional Fγ does not depend on γ if σ̃ =
−σ − n+1, because, first, we have formula (1), and,
second, f1 and f2 are both homogeneous functions,
and, third, the G-invariantmeasure on C can be rep-
resented in the form

dx =
dxζ(1) . . .dxζ(n)

|xζ(n+1)|
, (2)

where ζ ∈ S
n
and S

n+1
is the permutation group of the

set {1, . . . , n +1}.
Let f ∈ Dσ and y ∈ H(1). We refer to the inte-

gral transform

Π(f)(y) :=Fγ(q̂
−σ−n+1(y, x), f)

as the Poisson transform [1].

2 Formulas related to sphere
and paraboloid

Let γ1 be the intersectionof the cone C and theplane
x0 = 1. Each point x ∈ γ1 depends on spherical pa-
rameters φ1, . . . , φn−1 by the formula

xs =
n−s∏
i=1

sinφi · cosφn−s+1, s �=0,

The research presented in this paper was supported by grant NK 586P-30 from the Ministry of Education and Science of the
Russian Federation.

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Acta Polytechnica Vol. 51 No. 1/2011

if angle φn−s+1 exists. Here φn−1 ∈ [0;2π) and
φ1, . . . , φn−2 ∈ [0;π).
The subgroup H1 � SO(n) acts transitively on

γ1, and any permutate ζ ∈ S
n+1

defines the H1-

invariant measure

dγ1 =
dγζ(2) . . .dγζ(n)

|xζ(n+1)|
.

The invariant measure in spherical coordinates is
given by 9.1.1.(9) [2]
Let γ2 be the intersection of cone C and the hy-

perplane x0+xn =1. Wedescribe everypoint x ∈ γ2
by the coordinates r, φ1, . . . , φn−2 according to the
formulas

x0 =
1+ r2

2
, xn =

1− r2
2

,

xs = r
n−s−1∏

i=1

sinφi cosφn−s, s /∈ {0, n}

(if angle φn−s exists), where r ≥ 0, φn−2 ∈ [0;2π)
and φ1, . . . , φn−3 ∈ [0;π).
We denote as H2 the subgroup of G acting tran-

sitively on γ2. H2 consists of the matrices

n(b)=

⎛
⎜⎜⎜⎜⎝
diag(1, . . . ,1︸ ︷︷ ︸

n−1

) bT bT

−b 1− b∗ −b∗

b b∗ b∗

⎞
⎟⎟⎟⎟⎠ ,

where b =(b1, . . . , bn−1) and b
∗ =
1
2
(b21+ . . .+b

2
n−1).

It is not too hard to derive the H2-invariantmea-
sure

dγ = rn−2dr
n−2∏
i=1

sinn−i−2 φi dφi

on γ2.
Let λ > 0, μ ∈ R, k0 ≥ k1 ≥ . . . ≥ kn−2 ≥ 0,

l1 ≥ . . . ≥ ln−2 ≥ 0, m1 ≥ . . . ≥ mn−2 ≥ 0, K =
(k0, k1, . . . , kn−3, ±kn−2), L = (l1, . . . , ln−3, ±ln−2),
M =(m1, . . . , mn−3, ±mn−2).
We will now deal with two bases in Dσ. One of

them consists of the functions

f σ1K (x)= x
σ−k0
0 Ξ

n
K(x),

where K = (k0, k1, . . . , kn−3, ±kn−2) ∈ Zn−1, ki ≥
ki+1 ≥ 0 and

ΞnT(x) =
n−3∏
i=1

r
ti−ti+1
n−i ·

C
n−i
2 −1

ti−ti+1

(
xn−i
rn−i

)
(x2 ± ix1)tn−2.

The second basis consists of the functions

f σ2(L,λ)(x) = (x0 + xn)
σ+ n−32 ·

(
λ

2

)l1 (λrn−1
2

)3−n
2 −l1

·

Jl1+n−32

(
λrn−1

x0 + xn

)
Ξn−1L (x),

where r2j = x
2
1+ . . .+x

2
j, L =(l1, . . . , ln−3, ±ln−2) ∈

Z
n−2, λ ≥ 0 and li ≥ li+1 ≥ 0. Suppose, in addition,
that the functions of the above bases are equipped
with the normalizing factors defined by formulas [2,
9.4.1.7, 10.3.4.9].
Let us consider the distribution

f σ1K (x)=
∑
L

∫ +∞
0

cσ12K,(L,λ) f
σ2
(L,λ)dλ. (3)

From the orthogonality of the functions ΞnT , we ob-
tain the property

Fγ(f
σ1
K , f

−σ−n−1,1
−K̃

)= δKK̃ .

From this property, it immediately follows that

cσ12K,(L,λ) =Fγ(f
σ1
K , f

−σ−n−1,2
(L,λ) ).

Let γ = γ1. Then from the formula∫ 1
−1
(1− x2)ν−

1
2 Cνm(x)C

ν
n(x)dx =0,

where m �= n, re ν > −
1
2
, we derive

Lemma 1. If
n−2∑
i=1

(ki − li)2 �=0, then cσ12K,(L,λ) =0.

Let us assume another situation.

Lemma 2. If
n−2∑
i=1

(ki − li)2 =0, then

cσ12K,(L,λ) =2
−σ+n+3k1−3 π−1 ik1 (n +2k0 −2)

1
2 ·

√
(k0 − k1)!λk1 Γ

(
n −1
2

)
Γ

(
n −1
2
+ k1

)
·

Γ
(n
2
+ k1 −1

)
Γ
1
2

(
n −1
2

)
Γ−1(n +2k1 − 2) ·

Γ−
1
2(n + k0 + k1 −2)

k0−k1∑
m=0

(−1)m (m!)−1 ·

Γ(n + k0 + k1 + m −2)Γ−1
(

n −1
2
+ k1 + m

)
·

Γ−1(k0 − k1 − m −1)Γ−1(−σ + k1 + m) ·

G2113

⎛
⎝λ2
4

∣∣∣∣∣∣
−m
−σ + k1 −1,

n −3
2
+ k1

⎞
⎠ .

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Acta Polytechnica Vol. 51 No. 1/2011

Proof. Suppose γ = γ2. Then we obtain the inte-
gral ∫ +∞

0
r

n−1
2 +l1 (r2 +1)σ−k1 ·

C
n
2 −k1−1

k0−k1

(
1− r2
1+ r2

)
J n−3

2 +l1
(λr)dr,

which can be solved explicitly after replacing

rk Jk(λr) =2
k λ−k G1002

((
λr

2

)2 ∣∣∣∣∣ 0k,0
)

according to formulas [3, 8.932.1, 8.932.2] and [4,
20.5.4].�

Theorem 1.

P
− n2+1
−σ− n2

(coshα)= 22n−
9
2 π−

3
2
√

n −1 ·

sinh
n
2 −1 α e(σ+n−1)α Γ

(n
2
−1
)
Γ

(
n +1
2

)
·

Γ−1(−σ)Γ−
1
2(n −1)

∫ +∞
0

λ−n+3 ·

G2113

⎛
⎝λ2
4

∣∣∣∣∣∣
0

−σ −1,0,
n −3
2

⎞
⎠ ·

G2113

⎛
⎝(λe−α)2

4

∣∣∣∣∣∣
0

σ −
n −1
2

,0,
n −3
2

⎞
⎠ dλ.

Proof. Suppose that the condition k1 =
l1, . . . , kn−2 = ln−2 holds. From the distribution (3),
we obtain

Π(f σ1K )=
∫ +∞
0

cσ12K,(L,λ)Π(f
σ2
(L,λ))dλ.

FurtherweassumeΠ(f σ1K )=Fγ1(q̂
−σ−n+1(y, x), f σ1K )

and Π(f σ2(L,λ)) = Fγ2(q̂
−σ−n+1(y, x), f σ2(L,λ)), then for

the case y = (coshα,0, . . . ,0,sinhα) and put K =
(0, . . . ,0).�

Consider the case SO(2,1) of the group SO(n,1).
In this case, K ≡ k and (L, λ) ≡ λ. The following
theorem is related to this case.

Theorem 2. If −1 < re σ < 0 and α �=0, then

P
−l+12
σ+12

(coshα)= (−1)l−12−σ−
l
2−

9
4 π−

1
2 ×

e−α sin(−πσ) sinhl+
1
2 α ·(

coshα +1
coshα −1

) l
2+

1
4

Γ(σ − l +1)Γ
(

l −
3
2

)
·

Γ−1
(

l +
1
2

) ∫ ∞
0

ρ−σ−1 Kσ+1(ρe
−α) · (4)

∞∑
s=0

(−1)n Γ−2(s +1)Γ−1(s − σ) ·

G2113

(
ρ2

4

∣∣∣∣∣ −s−σ −1, 0
)
dρ

Proof. After repeating the proof of the previous
theorem, we derive the following representation of
the Gauss hypergeometric function:

2F1

(
−σ −

1
2
, σ +

3
2
;
1
2
+ l;
1−coshα
2

)
=

(−1)l−12−σ−
5
2 π−

1
2 e−α sinhα sin(−πσ) ·(

coshα +1
coshα −1

) l
2+

1
4

Γ(σ +1− l)Γ
(

l −
3
2

)
·∫ ∞

0
λ−σ−1 Kσ+1(λe

−α)
∞∑

s=0

(−1)n Γ−2(s +1) ·

Γ−1(s − σ)G2113

(
λ2

4

∣∣∣∣∣ −s−σ −1, 0, 0
)
dλ.

Now we use the formula [5, 7.3.1.88] for l =0.�

3 Formulas related to
paraboloid and hyperboloid

Let γ3+ be the intersection of cone C and the plane
xn =1. We denote as γ3− the intersection of C and
the plane xn = −1. Let γ3 := γ3+ ∪ γ3−. The con-
tour γ3 is a homogeneous space with respect to the
subgroup H3 � SO(n−1,1). If x belongs to γ3, then

xn = ±1, x0 =cosht,

xs = sinht
n−s−1∏

i=1

sinφi ·cosφn−s, s /∈ {0, n}

(if angle φn−s exists), where t ∈ R, φn−2 ∈ [0;2π)
and φ1, . . . , φn−3 ∈ [0;π).
Any permutation ζ ∈ S

n
determines the H3-

invariantmeasure

dγ4 =
dxζ(1) . . .dxζ(n−1)

|xζ(n)|

on γ3, so

dγ3 =cosh
n−2 tdt

n−2∏
i=1

sinn−i−2 φi dφi.

Let us now consider the basis consisting of the
functions

f σ2(M,μ,±)(x) = (xn)
σ+ n−32
± r

3−n
2 −m1

n−1 ·

P
3−n
2 −m1

−12+iμ

(
x0
xn

)
Ξn−1M (x),

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Acta Polytechnica Vol. 51 No. 1/2011

where (xn)
σ+ n−32
± is the generalized function defined

as

(xn)
σ+ n−32
± =

{
|xn|σ+

n−3
2 , if sign xn = ±1,

0, if sign x �= ±1,

M = (m1, . . . , mn−3, ±mn−2) ∈ Zn−2, mi ≥ mi+1 ≥
0 and μ ∈ R.
By analogywith the previous case, we can obtain

the coefficients cK,(M,μ,+). Let us suppose that n =3
and K =(l, s), M ≡ m. From the distribution

f σ3m,μ,+(x)=
∞∑
l=0

|l|∑
s=−|l|

cl,s,m,μ,+ f
σ1
l,s(x),

we have

f σ3−s,μ,+(x)=
∞∑
l=0

cl,s,−s,μ,+ f
σ1
l,s(x)

and, therefore,

Π(f σ3−s,μ,+)=
∞∑
l=0

|l|∑
s=−|l|

cl,s,−s,μ,+Π(f
σ1
l,s). (5)

We choose γ3 (in fact, γ3+) on the left side of equality
(5) and γ1 on the opposite side. In accordance with
our choice, we use two parametrizations of a point
y ∈ H(1):

y(v)=

(
v + v−1

2
,0, . . . ,0,

v−1 − v
2

)
and y(t) = (cosht,0, . . . ,0,sinht) respectively, so
v = e−t. After integration we have

sin[π(σ +1)] cosh−1 tΓ

(
iμ − σ −

1
2

)
·

Γ

(
−
3
2
− σ − iμ

)
P σ+1
−12+iμ

(tanht)=

√
2π

3
2

∞∑
l=0

(−1)l (l!)−1 Al sinh
1
2 t ·

Γ(l +1)Γ−1(σ − l +1)P −
1
2 −l

σ+12
(cosht),

where Al is the normalizing factor of the function
f σ1l,s(x).

References

[1] Vilenkin, N. Ja., Klimyk, A. U.: Representation
of Lie groups and special functions, Vol. 2, 1993.

[2] Vilenkin, N. Ja.: Special functions and theory of
group representations, 1968.

[3] Erdelyi, A.: Tables of integral transforms, 1954.

[4] Gradstein, I. S., Ryshik, I. M.: Tables of series,
products and integrals, 1981.

[5] Prudnikov, A. P., Brychkov, Yu. A., Mari-
chev, O. I.: Integrals and Series, Vol. 3: More
Special Functions, 1989.

Ilya Shilin
Dept of Higher Mathematics
M. Scholokhov Moscow State
University for the Humanities
Verhnya Radishevskaya 16–18
Moscow 109240, Russia
Dept 311
Moscow Aviation Institute
Volokolamskoe shosse 4, Moscow 125993, Russia

Aleksandr Nizhnikov
Moscow Pedagogical State University
M. Pirogovskaya 1, Moscow 119991, Russia

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