Table Of ContentLaplacian eigenmodes for the three sphere
M. Lachi`eze-Rey
Service d’Astrophysique, C. E. Saclay
4 91191 Gif sur Yvette cedex, France
0
0
February 1, 2008
2
n
a
J Abstract
4 The vector space k of the eigenfunctions of the Laplacian on the
1 three sphere S3, correVsponding to the same eigenvalue λ = k (k +
k
2), has dimension (k +1)2. After recalling the standard bases−for k,
P] we introduce a new basis B3, constructed from the reductions to S3Vof
S peculiarhomogeneousharmonicpolynomiainvolvingnullvectors. Wegive
. thetransformation laws between this basis and theusualhyper-spherical
h
harmonics.
at Thanks to the quaternionic representations of S3 and SO(4), we are
m able to write explicitely the transformation properties of B3, and thus
of any eigenmode, under an arbitrary rotation of SO(4). This offers the
[
possibility to select those functions of k which remain invariant under
V
1 a chosen rotation of SO(4). When the rotation is an holonomy transfor-
v mation of a spherical space S3/Γ, this gives a method to calculates the
3 eigenmodes of S3/Γ, which remains an open probleme in general. We
5
illustrate our method by (re-)deriving the eigenmodes of lens and prism
1
space. Inaforthcomingpaper,wepresentthederivationfordodecahedral
1
space.
0
4
0 1 Introduction
/
h
t TheeigenvaluesoftheLaplacian∆ofS3 areoftheformλ = k(k+2),
ma where k IN+. For a given value of k, they span the eigkensp−ace k of
dimension∈(k+1)2. Thisvectorspaceconstitutesthe(k+1)2 dimensVional
:
v irreductible representation of SO(4), theisometry group of S3.
i There are two commonly used bases (hereafter B1 and B2) for k
X V
which generalize in some sense (see below) the usualspherical harmonics
r Y for the two-sphere. The functions of these bases have a friendly be-
a ℓm
haviorundersomeoftherotationsofSO(4);thisgeneralizestheproperty
oftheY tobeeigenfunctionsoftheangularmomentumoperatorinIR3.
ℓm
However, these functions show no peculiar properties under the general
rotation of SO(4).
Exceptedforsomecases(lensandprismspaces,seebelow),thesearch
for the eigenmodes of the spherical spaces of the form S3/Γ remains an
open problem. Since those are eigenmodes of S3 which remain invariant
under the rotations of Γ, it is clear that this search requires an under-
standingof therotation properties of thebasis functions underSO(4).
The task of this paper is to examine the rotation properties of the
eigenfunctions of k, as a preparatory work for the search for eigenfunc-
V
1
tions of S3/Γ (in particular for dodecahedral space). This will be done
through the introduction of a new basis B3 of k (in the case k even),
V
for which the rotation properties can be explicitely calculated: following
a new procedure (that was already applied to S2 in [5]) we generate a
system of coherent states on k. We extract from it a basis B3 of k,
V V
which seems to have been ignored in theliterature, and presents original
properties. EachfunctionΦk ofthisbasisB3isdefinedas[thereduction
IJ
to S3 of] an homogeneous harmonic polynomial in IR4, which takes the
very simple form (X N)k. Here, the dot product extends the Euclidean
[scalar]dotproductof·IR4toitscomplexificationC4,andN isanullvector
ofC4,thatwespecifybelow. Afterdefiningthesefunctions,weshowthat
theyformabasisof k,andwegivetheexplicittransformation formulae
V
between B2 and B3.
ThepropertiesofthebasisB3differfromthoseofthetwootherbases,
and makeit more convenientfor particular applications. In particular, it
is possible to calculate explicitely its rotation properties, under an arbi-
traryrotationofSO(4),byusingtheirquaternionicrepresentation(section
3). This allows to find those functions which remain invariant under an
arbitrary rotation. In section 4, we apply these result to (re-)derive the
eigenmodes of lens and prism space.
2 Harmonic functions
A function f on S3 is an eigenmode [of the Laplacian] if it verifies ∆f =
λf. Itisknownthateigenvaluesareoftheformλ = k(k+2),k IN+.
k
The corresponding eigenfunctions generate the eigen[−vector]space∈k, of
V
dimension(k+1)2,whichrealizesanirreducibleunitaryrepresentationof
thegroup SO(4).
First basis
I call B1 the most widely used basis for k provided by the hyper-
V
spherical harmonics
B1 ( Y ), ℓ=1..k, m= ℓ..ℓ. (1)
kℓm ℓm
≡ Y ∝ −
It generalizes the usual spherical harmonics Y on the sphere. In fact,
ℓm
it can be shown ([1], [2] p.240,[3]) that a basis of this type exists on any
sphere Sn. Moreover, [2] [3] show that the B1 basis for Sn is “ naturally
generated ” by the B1 basis for Sn−1. In this sense, the B1 basis for S3
is generated by theusual spherical harmonics Y on the2-sphere S2.
ℓm
The generation process involves harmonic polynomials constructed
from null complex vectors (see below). The basis B1 is in fact based on
thereduction of the representation of SO(4) to representations of SO(3):
each isaneigenfunctionofanSO(3)subgroupofSO(4)whichleaves
kℓm
Y
a selected point of S3 invariant. This make these functions useful when
oneconsiders theaction ofthat peculiarSO(3) subgroup. Buttheyshow
no simple behaviour under a general rotation. We will no more use this
basis.
Second basis
By group theoretical arguments, [1] construct a different ON basis of
Vk,which is specific toS3:
B2 (T ), m ,m = k/2...k/2, (2)
≡ k;m1,m2 1 2 −
where m and m vary independently by entire increments (and, thus,
1 2
take entire or semi-entire values according to the parity of k). In the
2
spirit of the construction refered above, B2 may be seen as generated
fromadifferentchoiceofsphericalharmonicsonS2. ThebasesB1andB2
appearrespectivelyadaptedtothesystemsofhypersphericalandtoroidal
(see below) coordinates to describe S3.
The formula (27) of [1], reduced to the three-sphere, shows that the
elementsofthisbasistakeaveryconvenientformifweusetoroidalcoordi-
nates(astheyarecalledby[7])onthethreesphereS3: (χ,θ,φ)spanning
the range 0 χ π/2, 0 θ 2π and 0 φ 2π. They are conve-
≤ ≤ ≤ ≤ ≤ ≤
nientlydefined(see[7]foramorecompletedescription)fromanisometric
embeddingof S3 in IR4 (as thehypersurface x IR4; x =1):
∈ | |
x0 = r cosχ cosθ
x1 = r sinχ cosφ
x2 = r sinχ sinφ
x3 = r cosχ sinθ
where (xµ), µ = 0,1,2,3, is a point of IR4. As shown in [7]), these
coordinates appear naturally associated to some isometries.
Very simple manipulations show that, with these coordinates, each
eigenfunction of B2 takes theform:
T (X) t (χ) eiℓθ eimφ, (3)
k;m1,m2 ≡ k;m1,m2
where the t (χ) are polynomials in cosχ and sinχ and we wrote,
k;m1,m2
for simplification, ℓ m +m , m m m .
1 2 2 1
≡ ≡ −
To have a convenient expression, we report this formula in the har-
monic equation expressed in coordinates χ,θ,φ. This leads to a second
orderdifferentialequation(cf. equ15of[7]). Thesolutionisproportional
to a Jacobi polynomial: t (χ) cosℓχ sinmχ Pm,ℓ(cos2χ), d
k;m1,m2 ∝ d ≡
k/2 m . Thus, we havethefinal expression for thebasis B2
2
−
T (X)=C [cosχ eiθ]ℓ [sinχ eiφ]m P(m,ℓ)[cos(2χ)], (4)
k;m1,m2 k;m1,m2 d
with C √(k+1) (k/2+m2)! (k/2−m2)! from normalization re-
k;m1,m2 ≡ π (k/2+m1)! (k/2−m1)!
quirements (the variation raqnges of m1 anf m2 imply that the quantities
underfactorial sign areentireand positive). Notealso theuseful propor-
tionality relations:
cosℓχ sinmχ P(m,ℓ) (cos2χ) cosℓχ sin−mχ P(−m,ℓ) (cos2χ)
k−ℓ−m) ∝ k−ℓ+m)
2 2
cos−ℓχ sinmχ P(m,−ℓ)(cos2χ).
∝ k+ℓ−m
2
The term ζm ξn eiℓθ eimφ in (4) defines the rotation properties of
≡
T under a specific subgroup of SO(4). This properties generalizes
k;m1,m2
the properties of the spherical harmonics on the two-sphere S2, to be
eigenfunctionsoftherotationoperatorP . Thisadvantagehasbeenused
x
by[7]tocalculate(fromaslightlydifferentbasis)theeigenmodesoflensor
prismspaces(seebelowsection4). However,theT havenosimple
k;m1,m2
rotation properties under the general rotation of SO(4). This motivates
thesearch for adifferent basis of k.
V
Note that the basis functions T have also been introduced in
k;m1,m2
[2] (p. 253), with their expression in Jacobi Polynomials. Note also that
theyarethecomplexcounterpartsofthoseproposedby[7](theirequ. 19)
to find the eigenmodes of lens and prism spaces. The variation range of
theindices m ,m here (equ. 2) is equivalentto their condition
1 2
ℓ + m k, ℓ+m=k, mod (2), (5)
| | | |≤
through thecorrespondence ℓ=m +m , m=m m .
1 2 2 1
−
3
2.1 Complex null vectors
A complex vector Z (Z0,Z1,Z2,Z3) is an element of C4. We extend
≡
theEuclideanscalarproductinIR4 tothecomplex(nonHermitian)inner
product Z Z′ Zµ (Z′)µ, µ=0,1,2,3. A null vector N is defined
as having z·ero≡normµN N NµNµ = 0 (in which case, it may be
P · ≡ µ
considered as a point on the isotropic cone in C4). It is well known that
P
polynomialsoftheform(X N)k,homogeneousofdegreek,areharmonic
·
if and only if N is a null vector. This results from
∆ (X N)k ∂ ∂ (X N)k =k (N N ) (X N)k−1 =0,
0 µ µ µ µ
· ≡ · ! ·
µ µ
X X
where ∆ is the Laplacian of IR4. Thus, the restrictions of such polyno-
0
mials belong to k. As we mentioned above, peculiar null vectors have
V
been used in [2] and [3] togenerate thebases B1 and B2.
ToconstructathirdbasisB3,letusfirstdefineafamilyofnullvectors
N(a,b) (cosa, i sinb, i cosb, sina), (6)
≡
indexed by two angles a and b describing the unit circle (they define
coherent states in IR4).
The polynomial [X N(a,b)]k is harmonic and, thus, can be decom-
·
posed onto the basis B2. It is easy to check that, like the scalar product
X N(a,b), this polynomial depends on a and b only through the com-
bin·ations ei(θ−a) and ei(φ+b), with their conjugates. This implies that its
decomposition on B2 takesthe form
[X N(a,b)]k = P T (X) e−ia(m1+m2) eib(m2−m1),
· k;m1,m2 k;m1,m2
mX1,m2
(7)
where the coefficients P do not depend on a,b. Now we intend to
k;m1,m2
finda basis of k in theform of such polynomials.
V
2.2 An new basis
2.2.1 Roots of unity
To do so, we consider the (k+1)th complex roots of unity which are the
powers ρI of
2iπ 2π
ρ ek+1 cosα+isinα, α . (8)
≡ ≡ ≡ k+1
Werecallthefundamentalproperty,whichwillbewidelyusedthereafter:
k
ρnI =(k+1) δDirac, (9)
I
n=0
X
where theequallity in theDirac must betaken mod k+1.
In a given frame, we consider thefamily of nullvectors
N N(Iα,Jα)=(cosIα, i sinJα, i cosJα, sinIα), I,J =0..k
IJ
≡
(10)
and wedefinethefunctionsΦk : Φk (X) (X N )k. Wereport such
IJ IJ ≡ · IJ
a function in equ.(7) to obtain its development in thebasis B2. Then we
multiply both terms by ρI(m1+m2)−J(m2−m1). Making the summations
4
overI,J (eachvaryingfrom 0tok),andusing(9),weobtain,inthecase
where k is even (that we assume hereafter):
k
= 1 ρI(m1+m2)−J(m2−m1) Φk , (11)
Tk;m1,m2 (k+1)2 IJ
I,J=0
X
where P T .
Tk;m1,m2 ≡ k;m1,m2 k;m1,m2
This gives the decomposition of any T (and thus, of any har-
k;m1,m2
monic function) as a sum of the (k+1)2 polynomials Φk , providing the
IJ
newbasis of k:
V
B3 (Φk ), I,J =0..k (k even). (12)
≡ IJ
The coefficients P involved in the transformation are calculated
k;m1,m2
in Appendix A. We obtain easily the reciprocal formula expressing the
changeof basis:
k/2
Φk = ρ−I(m1+m2)+J(m2−m1). (13)
IJ Tk;m1,m2
m1,mX2=−k/2
4
3 Rotations in IR
3.1 Matrix representations
TheisometriesofS3 aretherotationsintheembeddingspaceIR4. Inthe
usualmatrixrepresentation,arotationisrepresentedbya4 4orthogonal
∗
matrix g SO(4),acting on the 4-vector(xµ) by matrix product.
Inthe∈complexmatrix representation,apoint(vector)ofIR3 isrepre-
sented bythe 2 2 complex matrix
∗
W iZ
X ; W x0+ix3, Z x1+ix2 C.
≡ iZ¯ W¯ ≡ ≡ ∈
(cid:20) (cid:21)
A rotation g is represented by two complex 2 2 matrices (G ,G ), so
L R
∗
thatitsaction takestheformX G X G (matrixproduct). Thetwo
L R
matricesG andG belongtoSU7→(2). SinceSU(2)identifieswithS3,any
L R
matrix G or G is of the same form than the matrix X above. Since
L R
SU(2) is also the set of unit norm quaternions, there is a quaternionic
representation for the action of SO(4).
3.2 Quaternionic notations
Letusnotej , µ=0,1,2,3thebasisofquaternions(thej correspondto
µ µ
theusual 1,i,j,k but we donot usethis notation here). We havej =1.
0
Ageneralquaternionisq=qµj =q0+qij (withsummationconvention;
µ i
the index i takes the values 1,2,3; the index µ takes the values 0,1,2,3).
Itsquaternionicconjugateisq¯ q0 qij . Thescalarproductisq q
i 1 2
(q q¯ +q q¯)/2, giving thequ≡atern−ionic norm q 2= qq¯ = (qµ)·2. ≡
1 2 2 1 | | 2 µ
Werepresentanypointx=(xµ)ofIR4 bythequaternionq xµ j .
Thepointsofthe(unit)sphereS3correspondtounitsquaternPioxns≡, q 2µ=
| |
1. Hereafter,allquaternionswillbeunitary(ifnootherwiseindicated). It
iseasytoseethat,usingthecoordinatesabove,apointofS3isrepresented
bythequaternion cosχ ζ˙+sinχ ξ˙j ,where we definedottedquantities,
1
like ζ˙ cosθ+j sinθ, ξ˙ cosφ+j sinφ, as the quaternionic analogs
3 3
≡ ≡
5
of the complex numbers ζ = cosθ+i sinθ and ξ = cosφ+i sinφ, i.e.,
with the imaginary i replaced by thequaternion j .
3
In quaternionic notation, the rotation g : x gx is represented by a
7→
pair of unit quaternions (Q ,Q ) such that q q =Q q Q .
L R x gx L x R
7→
Complex quaternions, null quaternions
The null vectors N introduced above do not belong to IR4 but to
C4. Thus, they cannot be represented by quaternions, but by complex
quaternions. Those are defined exactly like the usual quaternions, but
withcomplexratherthanrealcoefficients. Notethatthepureimaginaryi
doesnotcoincidewithanyofthej ,butcommuteswithallofthem. Also,
µ
complex conjugation (star) and quaternionic conjugation (bar) must be
carefully distinguished. Then it iseasy tosee that the(null) vectorsN
IJ
defined above correspond to the complex quaternion n ρ˙I +i j ρ˙J.
IJ 2
≡
Notethat n 2=0.
IJ
| |
In quaternionic notations, the basis functions are expressed as
n q¯ +q n¯ k
Φ (x)=(N x)k =<n q >k= IJ X X IJ . (14)
IJ IJ · IJ · X 2
(cid:16) (cid:17)
Quaternionic notations will help us to check how the basis functions are
transformed by therotations of SO(4).
3.3 Rotations of functions
To any rotation g, is associated its action R on functions: R : f
g g
7→
R f; R f(x) f(gx). Let usapply this action tothe basis functions:
g g
≡
Φ (x)=Φ (gx)=<n (Q q Q )>k . (15)
g IJ IJ IJ L x R
R ·
WeconsiderafunctiononS3 alsoasafunctionsonthesetofunitquater-
nions (q is theunit quaternion associated to the point x of S3). On the
x
otherhand, we may develop this function on thebasis:
k
R Φ Gij (g) Φ . (16)
g IJ ≡ IJ ij
ij=0
X
The coefficients Gij (g) of the development, that we intend to calculate,
IJ
completely encode the action of therotation g on the basis B3, and thus
on Vk.
To proceed , we introducethree auxiliary complex quaternions:
α 1+i j , β j i j =(1 i j ) j and δ j i j .
3 1 2 3 1 1 2
≡ ≡ − − ≡− −
They havezero norm and obey theproperties <α n >=ρI,
IJ
< α¯ n >= ρ−I, < β n >= ρJ, < δ n ·>= ρ−J. Let us now
IJ IJ IJ
· · ·
estimatetherelation(16)forthespecificquaternionα+Rα¯+S β+T δ,
with R,S,T arbitrary real numbers:
( +R ′+S +T )k = Gij (g) <(ρi+Rρ−i+Sρj+T ρ−j)k, (17)
A A B D IJ
ij
X
where <Q αQ n >, ′ <Q α¯ Q n >, <Q β Q
L R IJ L R IJ L R
A≡ · A ≡ · B≡ ·
n >, <Q δQ n >characterizetherotation. (Notethatthese
IJ L R IJ
D≡ ·
quantitiesdepend on I and J).
Wedevelop and identify thepowers of the exponentsR,S,T:
q ′p−q r k−p−r = Gij (g) ρi(2q−p) ρj(2r−k+p).
A A B D IJ
ij
X
6
This holds for 0 q p, 0 r k p, 0 p k. After definition of
≤ ≤ ≤ ≤ − ≤ ≤
thenewindices A q+r, B q r+k p,which both vary from 0to
≡ ≡ − −
k, theprevious equation takes theform
A/2 B/2 ′ p/2 ′ k/2
AB AD AA ABD = Gij (g)ρi(A+B−k)+j(A−B).
′ ′ IJ
(cid:18)AD(cid:19) (cid:18)AB(cid:19) (cid:18) BD (cid:19) (cid:18) A (cid:19) ij
X
This holds for any value of A,B,p. A consequence is that ′ = ,
AA BD
which can be checkeddirectly. Finally,
A B ′ k = Gij (g)ρi(A+B−k) ρj(A−B),
U V A IJ
ij
(cid:0) (cid:1) X
with B , A .
U ≡ A′ V ≡ B
Takingintoaccountthepropertiesof theroots ofunity,thisequation
(cid:0) (cid:1) (cid:0) (cid:1)
has thesolution
Gij = (A′)k k ρ−i(A+B−k) ρ−j(A−B) A B. (18)
IJ (k+1)2 U V
A,B=0
X
When a rotation is specified, there is no difficulty to estimate theassoci-
ated values of ′, , , and thus of these coefficients which completely
A U V
encode the transformation properties of the basis functions of Vk under
SO(4).
Inthenextsection, we applytheseresultstorederivetheeigenmodes
ofLensorPrismspace. Inthenextpaper[6],wetakeforgthegenerators
of Γ, the group of holonomies of the dodecahedral space. This will allow
theselection of the invariant functions, which constitute its eigenmodes.
4 Lens and Prism space
The eigen modes for Lens and Prism space havebeen found by [7]. Here
we derivethem again for illustration of our method.
4.1 Lens space
An holonomy transformation of a lens space takes the form, in complex
notation,
eiψ1+2ψ2 0 eiψ1−2ψ2 0
G = , G = G.
L " 0 e−iψ1+2ψ2 # R " 0 e−iψ1−2ψ2) #
(19)
Itsaction on a vector of IR4 takesthe form
W iZ Weiψ1 iZeiψ2
X ≡ iZ¯ W¯ 7→GL X GR = iZe−¯iψ2 W¯e−iψ1 . (20)
(cid:20) (cid:21) (cid:20) (cid:21)
In this simple case, W x0 +ix3, Z x1 +ix2 are transformed into
W eiψ1 and Z eiψ2 resp≡ectively. This≡corresponds to the quaternionic
notation
Q =w˙ w˙ , Q =w˙ /w˙ , w˙ cos(ψ /2)+j sin(ψ /2). (21)
L 1 2 R 1 2 i i 3 i
≡
The rotation is expressed in the simplest way in the toroidal coordi-
nates, since it acts as θ θ+ψ , φ φ+ψ . From the expression (4)
1 2
7→ 7→
of thebasis functions (B2), it result their transformation law :
R :T T eℓψ1+mψ2.
g k;m1,m2 7→ k;m1,m2
7
This leads directly to the invariance condition ℓψ +mψ = 0 mod 2π.
1 2
Usingthe standard notation for a lens space L(p,q), namely
ψ =2π/p, ψ =2π q/p,
1 2
we are led tothe conclusion:
theeigenmodes of lens space L(p,q) are all linear combinations of T ,
k;m1,m2
where theunderliningmeans that theindices verify thecondition
m +m +q(m m )=0, modulo(p).
1 2 2 1
−
4.2 Prism space
Thetwogenerators aresingle action rotations (G =0). Thefirst gener-
R
ator, analog to the lens case above, with ψ =ψ =2π/2P, provides the
1 2
first condition ℓ+m=0, mod 2P which takes theform
m =0 mod P. (22)
2
This implies that k must be even.
0 i
ThesecondgeneratorhasthecomplexmatrixformG=GL = i −0 ,
(cid:20) − (cid:21)
which corresponds to the quaternion Q = Q = j . Easy calculations
L 1
lead to = ρJ, ′ =ρ−J, =ρI, = ρ−I. R−eportingin (18) gives
A − A B D −
Gij = ρ(i−J) k k ρA(−i−j+I+J)+B (−i+j−I+J) ( 1)B. (23)
IJ (k+1)2 −
AX,B=0
Thisformula, togetherwith thoseexpressingthechangeofbasis between
B2andB3,allowtoreturntotherotationpropertiesofthebasisB2which
takethesimple form:
R:Tk;m1,m2 7→(−1)m2+k/2 Tk;m1,−m2. (24)
Itresultsimmediately thattheG-invariantfunctionsarecombinations of
Tk;m1,m2 +(−1)m2+k/2 Tk;m1,−m2.
Finally,
theeigenfunctions of thePrism space are combinations of
Tk;m1,m2 +(−1)m2+k/2 Tk;m1,−m2, ∀m1; k even.
Accordingtotheparity of k/2, thefunctions areincluded ornot,
Tk;m1,0
from which simple counting give themultiplicity as
(k+1) (1+[k/2P]), for k even ([...] means entire value),
(k+1) [k/2P], for k odd, in accordance with [4].
5 Conclusion
WehaveshownthatVk,thespaceofeigenfunctionsoftheLaplacianofS3
with a given eigenvalue λ (k even) admits a new basis B3. In contrary
k
to standard bases (B1 and B2) which show specific rotation properties
under selected subgroups of SO(4), it is possible to calculate explicitely
the rotation properties of B3 under any rotation of the group SO(4),
as well as to calculate the functions invariant under this rotation. This
opens the door to the calculation of eigenmodes of spherical space. The
8
eigenfunctions of lens and prism spaces had been calculated by [7], by
using a basis related to B2 (its real, rather than complex, version). We
rederived them to illustrate theproperties of thebases.
In a subsequent paper [6], we apply these results to the search of
the eigenfunctions of the dodecahedral space S3/Γ, where Γ = D∗ is
P
the binary dihedral group of order 4P. Those functions, still presently
unknown, are the eigenfunctions of S3 which remain invariant under the
elements of Γ.
5.1 Appendix A
Let us evaluate thefunction
k
Zk (X) ρℓI−Jm Φk (X) (25)
ℓm ≡ IJ
IXJ=0
=2−k ρℓI−mJ cosχ (ζρ−I+ 1 )+sinχ (ξρJ 1 ) k,
ζρ−I − ξρJ
XIJ (cid:20) (cid:21)
where we defined ζ eiθ and ξ eiφ. After development of the power
≡ ≡
with the binomial coefficients, thesum becomes
k
ρℓI−mJ k [cosχ (ζρ−I+ 1 )]k−p [sinχ (ξρJ 1 )]p. (26)
p ζρ−I − ξρJ
XIJ Xp=0(cid:18) (cid:19)
Let us write the identities
k−p
ρℓI (ζρ−I+ 1 )k−p = k−p ζ2r+p−k ρ−I(2r+p−k−ℓ), (27)
ζρ−I r
r=0 (cid:18) (cid:19)
X
p
ρ−mJ (ξρJ 1 )p = p ξ2q−p( 1)p−q ρJ(2q−p−m), (28)
− ξρJ q −
q=0(cid:18) (cid:19)
X
that we insert into (26). After summing over I,J, and rearranging the
terms, we obtain:
Zℓm(X)=2−k ζℓ ξm k! k q(!−(q1)q−mm)!(c(oks+χℓ−)k2−q+2qm+)m!((ksi−nℓχ−)22qq+−mm)! . (29)
Xq − 2 2
Thisformularesultsfromthefactthat,through(9),thesummationsover
I,J imply p = 2q m and 2r = ℓ+k+m 2q, that we have reported.
− −
Therange of thesummation over q is defined bytheconditions
0 ℓ+k+m 2q 2k+2m 4q 2k, 0 q 2q m k. (30)
≤ − ≤ − ≤ ≤ ≤ − ≤
Rearrangements of the previous formula, inserting u cos(2χ) =
2cos2χ 1=1 2sin2χ,lead to ≡
− −
2−3k/2 ζℓ ξm k! (1+u)2ℓ (1 u)m2
Zℓm(X)= (m+d)! (ℓ+d)! − (31)
m+d ℓ+d (1+u)i (u 1)d−i,
i d i −
q (cid:18) (cid:19) (cid:18) − (cid:19)
X
9
wherewehavedefinedi k+m−ℓ q andd k−ℓ−m. Verificationshows
≡ 2 − ≡ 2
thattherangedefinedasabovegivesexactlythedevelopmentformulafor
theJacobi polynomial. The comparison with (11) gives the coefficient
2−k k!
P =
k;m1,m2 (k/2 m )! (k/2+m )! (k+1)2 C
− 1 1 k;m1,m2
2−k π k! (k+1)−5/2
=
(k/2+m )! (k/2 m )!(k/2+m )! (k/2 m )!
2 2 1 1
− −
p
References
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[2] Higher transcendental Functions A. Erd´elyi, W. Magnus, F. Ober-
hettinger, F. G. Tricomi, McGraw-Hill 1953
[3] A.Fryant,SIAMJ. Math. Anal., Vol. 22, N 1, pp.268-271, 1991
[4] IkedaA. 1995, Kodai Math . J. 18 (1995) 57-67
[5] Lachi`eze-Rey M. 2003, Journal of Physics A: Mathe-
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ration
[7] Lehoucq R.Uzan J. P., WeksJ. 2002, math.SP/0202072
[8] Fonctions sp´eciales de la physique math´ematique, A. Nikiforov, V.
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