Spherical Functions and OrthogonalPolynomials

Juan Tirao

Orthogonal Polynomials and Image ProcessingCarmona, Spain

October 15-17, 2007

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Outline

1. Spherical Harmonics

2. Zonal spherical functions

3. Spherical functions

4. Matrix hypergeometric equation

5. Spherical functions associated to P2(C)

6. Project

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Spherical Harmonics

A function f = f(x, y, z) is harmonic if

∂2f

∂x2+∂2f

∂y2+∂2f

∂z2= 0.

It is homogeneous of degree n if

f(λx, λy, λz) = λnf(x, y, z).

The homogeneous harmonic polynomials can be considered as functions onthe unit sphere S2 in R3. They are called spherical harmonics.

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Let (r, θ, φ) be ordinary polar coordinates in R3:

x = r sin θ cosφ, y = r sin θ sinφ, z = r cos θ.

In terms of these coordinates the Riemannian structure of R3 is

ds2 = dr2 + r2dθ2 + r2(sin θ)2dφ2,

and the Laplace operator is

∆ =∂2

∂r2+

1r2∂2

∂θ2+

1r2(sin θ)2

∂2

∂φ2+

2r

∂

∂r+

cos θr2 sin θ

∂

∂θ.

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If f = f(θ) is a spherical harmonic of degree `, then

d2f

d θ2+

cos θsin θ

df

dθ+ `(`+ 1)f = 0.

By making the change of variables y = (1 + cos θ)/2 we get

y(1− y)d2f

d y2+ (1− 2y)

df

dy+ `(`+ 1)f = 0.

The bounded solution at y = 0, up to a constant, is 2F1(−`, ` + 1, 1; y).Since the Legendre polynomial of degree ` is given by

P`(x) = 2F1

(−` , `+ 1

1 ; (1 + x)/2),

we get that f(θ) = P`(cos θ)f(0).

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Let o = (0, 0, 1) be the north pole of S2, and let φ(p) = P`(cos(θ(p))) forp ∈ S2. Then φ is the unique spherical harmonic of degree `, constantalong the parallels and such that φ(o) = 1.

The set V` of all complex linear combinations of translates φg(p) = φ(g · p),g ∈ SO(3), is the linear space of all spherical harmonics of degree `.

The action of SO(3) in V` is an irreducible representation of SO(3) ofdimension 2`+ 1, and these are all.

Moreover we have the following unitary direct sum

L2(S2) =∑

`

V`.

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Legendre and Laplace found that the Legendre polynomials satisfy thefollowing addition formula

P`(cosα cosβ + sinα sinβ cosφ)

= P`(cosα)Pn(cosβ) + 2∑k=1

(`− k)!(`+ k)!

P k` (cosα)P k

` (cosβ) cos kφ,(1)

where the P k` ’s are the associated Legendre polynomials.

By integrating (1) we get

P`(cosα)P`(cosβ) =12π

∫ 2π

0

P`(cosα cosβ + sinα sinβ cosφ)dφ. (2)

Moreover the Legendre polynomials can be determined as solutions to (2).This integral equation can now be expressed in terms of the function φ

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on SO(3) defined by φ(g) = φ(g · o) = P`(cos(d(o, g · o))). In fact (2) isequivalent to

φ(g)φ(h) =∫

K

φ(gkh) dk, (3)

where K denotes the compact subgroup of SO(3) of all elements which fixthe north pole o, and dk denotes the normalized Haar measure of K.

In fact, let A denote the subgroup of all elements of SO(3) which fixthe point (0, 1, 0). Then SO(3) = KAK. Thus to prove () it is enough toconsider rotations g and h around the y-axis through the angles α and β,respectively. Then if k denotes the rotation of angle φ around the z-axis wehave

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gkh · o =

(− cosα cosφ sinβ+sinα cosβ,− sinφ sinβ, sinα cosφ sinβ+cosα cosβ)t.

Thus cos(θ(g · o)) = cosα cosβ + sinα sinβ cosφ and

φ(gkh) = P`(cosα cosβ + sinα sinβ cosφ).

Then equation (3) becomes (2).

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We have ∫S2φm(p)φn(p) dp = 0

dp = sin θ dθ dφ. If m 6= n, then

∫ 2π

0

∫ π

0

Pm(cos θ)Pn(cos θ) sin θ dθ dφ = 0

∫ π

0

Pm(cos θ)Pn(cos θ) sin θ dθ =∫ 1

−1

Pm(x)Pn(x) dx = 0

Lagrange(2`+ 1)xp`(x) = (`+ 1)p`+1(x) + `p`−1(x)

p−1(x) = 0, p0(x) = 1

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The three term recursion relation revisited:

ρ : SU(2) → SO(3) covering isomorphism

V` becomes a SU(2)-module. It has a unique orthonormal basis {vi}2`0 :

H =(

1 00 −1

), E =

(0 10 0

), F =

(0 10 0

);

Hvi = 2(`− i)vi, Evi = ivi−1, Fvi = (2`− i)vi+1.

Up to a constant φ` = v`.

V` ⊗ V1 = V`+1 ⊕ V` ⊕ V`−1, v` ⊗ v1 = a1v`+1`+1 + a2v

`−1`−1,

Clebsch-Gordan coefficients

|a1|2 =`+ 12`+ 1

, |a2|2 =`

2`+ 1.

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Nowφ`(g) = 〈g · v`, v`〉

In fact φ(g) = 〈g · v`, v`〉 ∈ V` by Schur’s orthogonality relations, it isK-invariant and φ(e) = 1.

〈g · v`, v`〉〈g · v1, v1〉 = 〈g · (v` ⊗ v1), v` ⊗ v1〉,

φ`(g)φ1(g) = 〈g · (a1v`+1`+1 + a2v

`−1`−1), a1v

`+1`+1 + a2v

`−1`−1〉

= |a1|2φ`+1 + |a2|2φ`−1,

which is equivalent to

(2`+ 1)xp`(x) = (`+ 1)p`+1(x) + `p`−1(x).

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Zonal Spherical Functions

Elie Cartan and Herman Weyl: let G be a locally compact group and K acompact subgroup.

φ(g)φ(h) =∫

K

φ(gkh) dk.

Compact Two Point Homogeneous Manifolds

Sk = SO(k + 1)/SO(k)P k(R) = SO(k + 1)/O(k)P k(C) = SU(k + 1)/U(k)

P k(H) = Sp(k + 1)/Sp(k)× Sp(1)P 2(Cay) = F4/Spin(9)

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Sk: p(α,α)n (cos θ) α = k−2

2

P k(R): p(α,β)n (cos θ) (α, β) = (k−2

2 ,−12)

P k(C): p(α,β)n (cos θ) (α, β) = (k − 1, 0)

P k(H): p(α,β)n (cos θ) (α, β) = (2k − 1, 1)

P 2(Cay): p(α,β)n (cos θ) (α, β) = (7, 3)

Rk: Bessel functions

hyperbolic spaces: Jacobi functions

Discrete two point homogeneous spaces

Sk × Zk2/Sk: Krawtchouk polynomials

S`/Sk × S`−k: Hahn polynomials

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Spherical Functions

G locally compact group, K compact subgroup, δ ∈ K

Φ : G −→ End(V ) continuous such that Φ(e) = I and

Φ(x)Φ(y) =∫

K

χδ(k−1)Φ(xky) dk

Lemma

(i) Φ(kgk′) = Φ(k)Φ(g)Φ(k′)

(ii) π = Φ|K is a representation of K and π ' nδ

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(E,U) irreducible representation of G in a Banach space E

E = ⊕k∈KE(δ), P (δ) =∫

K

χδ(k−1)U(k) dk

Theorem. If dimE(δ) <∞ then

Φ(g)a = P (δ)U(g)a, a ∈ E(δ)

is a quasi bounded irreducible spherical function of type δ. The

converse is also true.

Godement and Harish-Chandra (1952): φ(g) = tr(Φ(g))

T (1970, 1977), Gangolli and Varadarajan (1988)

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Let G be a Lie group, (V, π) representation of K, π ' nδ

Theorem

Φ : G −→ End(V ) continuous is spherical iff

i) Φ analytic, and Φ(e) = Iii) Φ(k1gk2) = π(k1)Φ(g)π(k2)iii) [DΦ](g) = Φ(g)[DΦ](e), D ∈ D(G)K

Moreover,D 7→ [DΦ](e)

is a representation of D(G)K which caracterizes Φ up to equivalence

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Matrix Hypergeometric Equation

A,B,C ∈ End(V ).

z(1− z)F ′′(z) + (C − z(A+B + I))F ′(z)−ABF (z) = 0.

Let

2F1

(A ; B

C; z

)=

∞∑m=0

zm

m!(C;A;B)m,

where the symbol (C;A;B)m is defined inductively by

(C;A;B)0 = 1,

(C;A;B)m+1 = (C +m)−1(A+m)(B +m)(C;A;B)m, m ≥ 0.

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Theorem (T, PNAS 2003) If no eigenvalue of C is an integer less or equalto zero, then 2F1

(A ; B

C; z

)is analytic on |z| < 1, with values in End(V ).

If F0 ∈ V thenF (z) = 2F1

(A ; B

C; z

)F0

is the unique analytic solution at z = 0 of the hypergeometric equationsuch that F (0) = F0.

More general hypergeometric equation. Given U, V,C ∈ End(V ).

z(1− z)F ′′(z) + (C − zU)F ′(z)− V F (z) = 0.

Let

2H1

(U ; V

C; z

)=

∞∑m=0

zm

m![C;U ;V ]m,

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where the symbol [C;U ;V ]m is defined inductively by

[C;U ;V ]0 = 1,

[C;U ;V ]m+1 = [C +m]−1(m2 +m(U − 1) + V )[C;U ;V ]m, m ≥ 0.

Theorem (T, PNAS 2003) If no eigenvalue of C is an integer less or equalto zero, then 2H1

(U ; V

C; z

)is analytic on |z| < 1, with values in End(V ).

If H0 ∈ V thenH(z) = 2H1

(U ; V

C; z

)H0

is the unique analytic solution at z = 0 of the hypergeometric equationsuch that H(0) = H0.

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Spherical functions associated to P2(C)

To express, as in the scalar case, all irreducible spherical functions, of agiven type, associated to P2(C) = SU(3)/U(2) in terms of the matrixhypergeometric function. Then, to build out of all these spherical functionsa sequence of matrix orthogonal polynomials.

G = SU(3), K = U(2), Φ : G −→ End(Vπ)

Φ(x)Φ(y) =∫

K

χπ(k−1)Φ(xky) dk, Φ(e) = I

Grunbaum, Pacharoni and T (JFA 2002); Roman and T (IJM 2006)

First examples of matrix orthogonal polynomials which are eigenfunctionsof a second order differential operator.

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D(G)K = D(G)G ⊗D(K)K.

In particular D(G)K is abelian, hence irreducible spherical functions are ofheight one.

1. Φ analytic,

2. Φ(k1gk2) = π(k1)Φ(g)π(k2),

3. [DΦ](g) = λDΦ(g), D ∈ D(G)G.

D(G)G = C[∆2,∆3] (∆2 is the Casimir operator of G)

π extends to a unique holomorphic representation of GL(2, C). Let A(g)be the 2× 2 upper left block of g, and let

A = {g ∈ G : detA(g) 6= 0}.

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Let Φπ(g) = π(A(g)), g ∈ A. To Φ we associate the function

H(g) = Φ(g) Φπ(g)−1, g ∈ G.

1. H(e) = I

2. H(gk) = H(g)

3. H(kg) = π(k)H(g)π(k−1)

4. H is an eigenfunction of certain matrix differential operators

H is a function on the affine space C2 ⊂ P2(C).

H is determined by its restriction to a cross section of the K-orbits in C2,which are the spheres of radios r ∈ [0,∞); H is a diagonal matrix or avector valued function.

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π(A) = πn,`(A) = (detA)nA`, n ∈ Z, ` ∈ Z≥0.

The irreducible spherical functions are orthogonal with respect to

〈Φ,Ψ〉 =∫

G

tr(Φ(g)Ψ(g)∗) dg

∆2 is a symmetric operator: 〈∆2Φ,Ψ〉 = 〈Φ,∆2Ψ〉.

Change of variables u = r2

1+r2, u ∈ [0, 1)

DH = u(1− u)H ′′ + (2− uA1)H ′ +1u

(B0 −B1 + uB1)H,

EH = u(1− u)MH ′′ + (C1 − C0 − uC1)H ′ +1u

(D0 +D1 − uD1)H.

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D is symmetric with respect to

〈H,K〉 =∫ 1

0

K∗(u)W (u)H(u) du

W (u) =∑0

u(1− u)n+`−iEi,i.

Theorem The irreducible spherical functions Φ of SU(3) of type(n, `) correspond precisely to the simultaneous C`+1-valued polynomialeigenfunctions H of the differential operators D and E, such thathi(u) = (1− u)i−n−`gi(u) for all n+ `+ 1 ≤ i ≤ ` with gi polynomial andH(0) = (1, . . . , 1)t.

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To put this into the framework of orthogonal polynomials we conjugate D:look for a function Ψ(u) such that D = Ψ(u)−1DΨ(u) be a hypergeometricoperator

D = u(1− u)d2

du2+ (C − uU)

d

du− V

The function ψ(u) = XT (u), where X is the Pascal matrix Xi,j =(

ij

)and

T (u) is the diagonal matrix T (u)i,i = ui, is such a function.

Theorem The irreducible spherical functions Φ of SU(3) of type (n, `) are ina one to one correspondence with the functions F (u) = ψ(u)−1H(u). Theseare precisely the simultaneous C`+1-valued polynomial eigenfunctions of thedifferential operatorsD and E = ψ−1Eψ, such that F (0) = (1, x1, . . . , x`)t.

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Therefore, if Φ is an irreducible spherical function on G of type (n, `), thenthere exist λ ∈ C and F0 = (1, x1, . . . , x`)t such that

F (u) = 2H1

(U ; V +λ

C;u

)F0.

Hence, there exits a nonnegative integer w such that [C;U ;V ]w+1 issingular and [C;U ;V ]w is not singular. This implies that

λ = −w(w + n+ `+ k + 2)− k(n+ k + 1), 0 ≤ k ≤ `.

LetWλ = {F = C`+1[u] : DF = λF}

Since F (0) determines F ∈Wλ, the linear map

ν : Wλ −→ C`+1, ν(F ) = F (0)

is a surjective isomorphism.

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The differential operators D and E commute, since they come,respectively, from ∆2 and ∆3 of D(G)G. Therefore E restricts to alinear operator of Wλ into itself. Then the following is a commutativediagram

WλE−→ Wλ

ν

y yν

C`+1 M(λ)−−−→ C`+1

M(λ) is the (`+ 1)× (`+ 1) matrix given by

M(λ) = Q0(C + 1)−1(U + V + λ)C−1(V + λ) + P0C−1(V + λ) +R.

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Theorem For any (n, `) and any λ ∈ C, the eigenvalues of M(λ) are

µj(λ) = λ(n− `+ 3j)− 3j(`− j + 1)(n+ j + 1),

0 ≤ j ≤ `, and all have geometric multiplicity one. Moreover, if (v0, . . . , v`)t

is a nonzero eigenvector of M(λ), then v0 6= 0.

Theorem If λ = λw,k = −w(w+n+`+k+2)−k(n+k+1), 0 ≤ k ≤ `,there exits a unique µ-eigenvector F0 = (1, x1, . . . , x`) of M(λ) such that

Fw,k(u) = 2H1

(U ; V +λ

C;u

)F0.

is a polynomial of degree w. Moreover µ = µk(λ).

LetPw = (Fw,0, . . . , Fw,`), w ≥ 0.

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Theorem {Pw} is a sequence of orthogonal polynomials with respect to theweight function ψ∗Wψ, and DPw = PwΛ(w), where Λw is the diagonalmatrix with Λ(w)k,k = λw,k.

The irreducible representations of SU(3) are labeled by m = (m1,m2,m3)with m1 ≥ m2 ≥ m3, integers. Let V be the standard representation.Assume that m1 ≥ n+ ` ≥ m2 ≥ n ≥ m3. Then by considering

Vm ⊗ V = Vm1 ⊕ Vm2 ⊕ Vm3,

m1 = (m1 + 1,m2,m3), m2 = (m1,m2 + 1,m3), m3 = (m1,m2,m3 + 1),

we obtain the three term recursion relation for the sequence {Pw}:

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34 Pacharoni and Tirao

H1(t; v, 0) = t!(n+3+v)/22F1

!(n+3+v)

2 ,(!n+3+v)

23

; 1! 1

t

",

H0(t; v, 1) = t!(n+4+v)/22F1

!(n+4+v)

2 ,(!n+2+v)

23

; 1! 1

t

",

H1(t; v, 1) = t!(n+2+v)/23F2

#(n+2+v)

2 ,(!n+2+v)

2 ,2(n+v+1)

n+v

3 ,n+2+v

n+v

; 1! 1

t

$.

The first row of the matrix function H(t; v) is H(t; v, 0) and the second one isH(t; v ! 1, 1). Then

H(t; v) =

!F11 F12

F21 F22

",

with

F11 = t!(n+3+v)/23F2

#(n+3+v)

2 ,(!n+1+v)

2 ,2(n!v)n+1!v

3 ,n!1!vn+1!v

; 1! 1

t

$,

F12 = t!(n+3+v)/22F1

!(n+3+v)

2 ,(!n+3+v)

23

; 1! 1

t

",

F21 = t!(n+3+v)/22F1

!(n+3+v)

2 ,(!n+1+v)

23

; 1! 1

t

",

F22 = t!(n+1+v)/23F2

#(n+1+v)

2 ,(!n+1+v)

2 ,2(n+v)n!1+v

3 ,n+1+vn!1+v

; 1! 1

t

$.

The matrices Av , Bv and Cv are

Av =

#(v+n+1)(v+n!3)(v!n!3)

4v(v!1)(v+n!1) 02(v+n!3)

(v!1)(v+n!1)(v!n!3)(v+n!3)(v+n!1)(v!n!5)

4(v!1)(v!2)(v!n!3)

$,

Bv =

#v4!n4!4n3+16n!6v2+21

2(v+1)(v!1)(v+n!1)(v!n!1)2(v!n!3)

v(v+n!1)(v!n!1)2(v!n!1)

(v!1)(v+n+1)(v!n!1)v4!4v3+8v+3+6n2!n4!8n2v(v!2)(v+n!1)(v!n!1)

$,

Cv =

#(v+n+3)(v!n!3)(v!n!3)

4v(v+1)(v!n+1)2(v+n+3)

v(v+n+1)(v!n!1)

0 (v+n!1)(v+n+3)(v!n!1)4v(v!1)(v+n+1)

$.

6.3. The general case. If we use infinite matrices the three term recursionrelation (46) can be written, for any w " C! 1

2Z , in the following way:

t

%%%%%%%%%%%%%%%%%%

···

hw!1

hw

hw+1

···

%%%%%%%%%%%%%%%%%%

=

%%%%%%%%%%%%%%%%%%

· · · · · · · · ·· · · · · · · · ·· · · · · · · · ·· 0 A"

w!1 B"w!1 C "

w!1 0 · · ·· · 0 A"

w B"w C "

w 0 · ·· · · 0 A"

w+1 B"w+1 C "

w+1 0 ·· · · · · · · · ·· · · · · · · · ·· · · · · · · · ·

%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%

···

hw!1

hw

hw+1

···

%%%%%%%%%%%%%%%%%%

. (48)

Pacharoni and Tirao 35

Also it is of interest to point out that if w = max{0,!n} then (46) impliesthat

u

&

'''''''''

P0

P1

P2

···

(

)))))))))

=

&

'''''''''

B0 C0 0 · · ·A1 B1 C1 0 · ·0 A2 B2 C2 0 ·· · · · · ·· · · · · ·· · · · · ·

(

)))))))))

&

'''''''''

P0

P1

P2

···

(

)))))))))

. (49)

This is precisely the three term recursion relation obtained in the case of thecomplex projective plane, see [14].

The coe!cient matrices appearing in (48) and (49) have the followinginteresting property: the sum of all the matrix elements in any row is equal toone. See the next proposition. Moreover all the entries of the coe!cient matrix in(49) are nonnegative real numbers. This may have important applications in themodeling of some stochastic phenomena.

Proposition 6.4. If 2w = !v! n! !! 2 and (v! k)(!! n! v! 2k) #= 0 for!1 $ k $ ! + 1 and !! n + v ! 2k #= 0 for 0 $ k $ 2! + 1, then

*

j

(A"w)i,j +

*

j

(B"w)i,j +

*

j

(C "w)i,j = 1. (50)

Proof. We refer the reader to (46). Then (50) is equivalent to,*

j

(Av)i,j +*

j

(Bv)i,j +*

j

(Cv)i,j = 1. (51)

Taking into account the definitions of Av, Bv and Cv given in (42) the aboveequation becomes

(Av)j,j!1 + (Av)j,j + (Bv)j,j!1 + (Bv)j,j + (Bv)j,j+1 + (Cv)j,j + (Cv)j,j+1 = 1. (52)

Using (41) and writing a1 = a1(r ! 3j, v ! j, j, !), a2 = a2(r ! 3j, v ! j, j, !) anda3 = a3(r ! 3j, v ! j, j, !) we have to check that

a1

+b2(r ! 3j + 1, v ! j ! 1, j, !) + b3(r ! 3j + 1, v ! j ! 1, j, !)

+ b1(r ! 3j + 1, v ! j ! 1, j, !),

+a2

+b2(r ! 3j ! 2, v ! j, j + 1, !) + b3(r ! 3j ! 2, v ! j, j + 1, !)

+ b1(r ! 3j ! 2, v ! j, j + 1, !),

+a3

+b2(r ! 3j + 1, v ! j + 1, j, !) + b3(r ! 3j + 1, v ! j + 1, j, !)

+ b1(r ! 3j + 1, v ! j + 1, j, !),

= 1,

(53)

where r = ! ! n . The proof will be completed once we established the followinglemma.

Lemma 6.5. If v " C and v(r ! v)(r + v) #= 0 then

i) a1(r, v, j, !) + a2(r, v, j, !) + a3(r, v, j, !) = 1,

ii) b1(r, v, j, !) + b2(r, v, j, !) + b3(r, v, j, !) = 1.

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Project

To express, as in the scalar case, all irreducible spherical functions ofa given type associated to a compact two point homogeneous space interms of a sequence of matrix orthogonal polynomials given by the matrixhypergeometric function.

First step: The complex projective space

G = SU(k + 1), K = U(k), G/K = P k(C)

Many Thanks to:

A. Duran, F.A. Grunbaum and I. Pacharoni, P. Roman from the UniversidadNacional de Cordoba, Argentina.

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