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Revista de la Unión Matemática Argentina
versão impressa ISSN 0041-6932versão On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.49 n.2 Bahía Blanca jul./dez. 2008
Matrix spherical functions and orthogonal polynomials: An instructive example
I. Pacharoni
This paper is partially supported by CONICET, FONCyT, Secyt-UNC and the ICTP.
Abstract. In the scalar case, it is well known that the zonal spherical functions of any compact Riemannian symmetric space of rank one can be expressed in terms of the Jacobi polynomials. The main purpose of this paper is to revisit the matrix valued spherical functions associated to the complex projective plane to exhibit the interplay among these functions, the matrix hypergeometric functions and the matrix orthogonal polynomials. We also obtain very explicit expressions for the entries of the spherical functions in the case of 2 × 2 matrices and exhibit a natural sequence of matrix orthogonal polynomials, beyond the group parameters.
The well known Legendre polynomials are a special case of spherical harmonics: the homogeneous harmonic polynomials of , considered as functions on the unit sphere
. Let
be ordinary polar coordinates in
:
,
and
In terms of these coordinates the Riemannian structure of
is given by the symmetric differential form
and the Laplace operator is
![2 2 2 Δ = -∂--+ 1--∂--+ ----1-----∂--+ 2-∂-+ -cosθ--∂--. ∂r2 r2∂θ2 r2(sin θ)2∂φ2 r∂r r2sinθ ∂θ](/img/revistas/ruma/v49n2/2a0210x.png)
![f](/img/revistas/ruma/v49n2/2a0211x.png)
![n](/img/revistas/ruma/v49n2/2a0212x.png)
![φ](/img/revistas/ruma/v49n2/2a0213x.png)
![2 d-f-+ cos-θdf-+ n(n + 1)f = 0. dθ2 sin θdθ](/img/revistas/ruma/v49n2/2a0214x.png)
![y = (1 + cos θ)∕2](/img/revistas/ruma/v49n2/2a0215x.png)
![2 y(1 - y)d-f-+ (1 - 2y )df-+ n(n + 1 )f = 0. d y2 dy](/img/revistas/ruma/v49n2/2a0216x.png)
![y = 0](/img/revistas/ruma/v49n2/2a0217x.png)
![2F1 (- n, n + 1,1;y)](/img/revistas/ruma/v49n2/2a0218x.png)
![n](/img/revistas/ruma/v49n2/2a0219x.png)
![( ) - n , n + 1 Pn (x) = 2F1 1 ;(1 + x)∕2 ,](/img/revistas/ruma/v49n2/2a0220x.png)
![f(θ) = Pn (cos θ)f(0)](/img/revistas/ruma/v49n2/2a0221x.png)
Let
![o = (0,0,1 )](/img/revistas/ruma/v49n2/2a0222x.png)
![S2](/img/revistas/ruma/v49n2/2a0223x.png)
![d(o,p)](/img/revistas/ruma/v49n2/2a0224x.png)
![p ∈ S2](/img/revistas/ruma/v49n2/2a0225x.png)
![o](/img/revistas/ruma/v49n2/2a0226x.png)
![φ(p) = Pn (cos(d (o,p )))](/img/revistas/ruma/v49n2/2a0227x.png)
![φ](/img/revistas/ruma/v49n2/2a0228x.png)
![n](/img/revistas/ruma/v49n2/2a0229x.png)
![φ(o) = 1](/img/revistas/ruma/v49n2/2a0230x.png)
![φg(p) = φ(g ⋅ p)](/img/revistas/ruma/v49n2/2a0231x.png)
![g ∈ SO (3)](/img/revistas/ruma/v49n2/2a0232x.png)
![n](/img/revistas/ruma/v49n2/2a0233x.png)
Legendre and Laplace found that the Legendre polynomials satisfy the following addition formula
![]() |
where the 's are the associated Legendre polynomials.
By integrating (1) we get
![]() |
Moreover the Legendre polynomials can be determined as solutions to (2). This integral equation can now be expressed in terms of the function on
defined by
. In fact (2) is equivalent to
![]() |
where denotes the compact subgroup of
of all elements which fix the north pole
, and
denotes the normalized Haar measure of
.
In fact, let denote the subgroup of all elements of
which fix the point
. Then
. Thus to prove (3) it is enough to consider rotations
and
around the
-axis through the angles
and
, respectively. Then if
denotes the rotation of angle
around the
-axis we have
![gkh ⋅o = (- cos α cos φ sin β+sin α cosβ, - sin φ sin β,sinα cosφ sinβ+cos α cosβ).](/img/revistas/ruma/v49n2/2a0258x.png)
![cos(d(o,g ⋅ o)) = cosα cosβ + sin α sin β cos φ](/img/revistas/ruma/v49n2/2a0259x.png)
![φ (gkh ) = Pn(cos α cosβ + sinα sinβ cosφ ).](/img/revistas/ruma/v49n2/2a0260x.png)
The functional equation (3) has been generalized to many different settings. One is the following. Let be a locally compact unimodular group and let
be a compact subgroup. A nontrivial complex valued continuous function
on
is a zonal spherical function if (3) holds for all
. Note that then
for all
and all
, and that
where
is the identity element of
.
The example above arises when
and
. The other compact connected rank one symmetric spaces have zonal spherical functions which are orthogonal polynomials in an appropriate variable. These polynomials are special cases of Jacobi polynomials and they can be given explicitly as hypergeometric functions.
The complex projective plane is another rank one symmetric space. In this case the zonal spherical functions are
.
A very fruitful generalization of the functional equation (3) is the following (see [T1] and [GV]. Let be a locally compact unimodular group and let
be a compact subgroup of
. Let
denote the set of all equivalence classes of complex finite dimensional irreducible representations of
; for each
, let
and
denote, respectively, the character and the dimension of any representation in the class
, and set
. We shall denote by
a finite dimensional complex vector space and by
the space of all linear transformations of
into
.
A spherical function on
of type
is a continuous function
such that
, (
= identity transformation) and
![∫ Φ (x)Φ (y) = χδ(k-1)Φ (xky) dk, for all x, y ∈ G. K](/img/revistas/ruma/v49n2/2a0297x.png)
When is the class of the trivial representation of
and
, the corresponding spherical functions are precisely the zonal spherical functions. From the definition it follows that
is a representation of
, equivalent to the direct sum of
representations in the class
, and that
for all
and all
. The number
is the height of
. The height and the type are uniquely determined by the spherical function.
2. Matrix valued spherical functions associated to
In [GPT1] the authors consider the problem of determining all irreducible spherical functions associated to the complex projective plane . This space can be realized as the homogeneous space
,
and
. In this case all irreducible spherical functions are of height one. Let
be any irreducible representation of
in the class
. Then an irreducible spherical function can be characterized as a function
such that
is analytic,
, for all
,
, and
,
, for all
and
.
Here denotes the algebra of all left and right invariant differential operators on
. In our case it is known that the algebra
is a polynomial algebra in two algebraically independent generators
and
, explicitly given in [GPT1].
The set can be identified with
in the following way: If
then
![π(k) = πn,ℓ(k ) = (det k)nkℓ,](/img/revistas/ruma/v49n2/2a02135x.png)
![ℓ k](/img/revistas/ruma/v49n2/2a02136x.png)
![ℓ](/img/revistas/ruma/v49n2/2a02137x.png)
![k](/img/revistas/ruma/v49n2/2a02138x.png)
For any we denote by
the left upper
block of
, and we consider the open dense subset
. Then
is left and right invariant under
. For any
we introduce the following function defined on
:
![Φπ(g) = π (A(g)),](/img/revistas/ruma/v49n2/2a02148x.png)
![π](/img/revistas/ruma/v49n2/2a02149x.png)
![GL (2, ℂ)](/img/revistas/ruma/v49n2/2a02150x.png)
![U(2)](/img/revistas/ruma/v49n2/2a02151x.png)
To determine all irreducible spherical functions of type
, we use the function
in the following way: in the open set
we define the function
by
![]() |
where is supposed to be a spherical function of type
. Then
satisfies
,
, for all
,
, for all
.
The canonical projection defined by
where
maps the open dense subset
onto the affine space
of those points in
whose last homogeneous coordinate is not zero. Then property ii) says that
may be considered as a function on
, and moreover from iii) it follows that
is determined by its restriction
to the cross section
of the
-orbits in
, which are the spheres of radius
centered at the origin. That is
is determined by the function
on the interval
. Let
be the closed subgroup of
of all diagonal matrices of the form
,
. Then
fixes all points
. Therefore iii) also implies that
for all
. Since any
as an
-module is multiplicity free, it follows that there exists a basis of
such that
is simultaneously represented by a diagonal matrix for all
. Thus, if
, we can identify
with a vector
![t ℓ+1 H (r) = (h0 (r),...,h ℓ(r)) ∈ ℂ .](/img/revistas/ruma/v49n2/2a02198x.png)
![Φ](/img/revistas/ruma/v49n2/2a02199x.png)
![Δ2](/img/revistas/ruma/v49n2/2a02200x.png)
![Δ3](/img/revistas/ruma/v49n2/2a02201x.png)
![H = H (r)](/img/revistas/ruma/v49n2/2a02202x.png)
![˜D](/img/revistas/ruma/v49n2/2a02203x.png)
![E˜](/img/revistas/ruma/v49n2/2a02204x.png)
![(0,∞ )](/img/revistas/ruma/v49n2/2a02205x.png)
Making the change of variables these operators become
If we denote by the
matrix with entry
equal to 1 and 0 elsewhere, then the coefficient matrices are
The following result, which characterizes the spherical functions associated to the complex projective plane is taken from Theorem 3.8 of [RT], see also [GPT1].
Theorem 2.1. The irreducible spherical functions of
of type
, correspond precisely to the simultaneous
-valued polynomial eigenfunctions
of the differential operators
and
, introduced in (5) and (6), such that
for all
with
polynomial and
.
We also obtain, from [GPT1] or [PT1], that there is a bijective correspondence between the equivalence classes of all irreducible spherical functions of type
and the set of pairs of integers
![]() |
Under this correspondence the function associated to the spherical function
satisfies
and
where
![]() |
2.1. Hypergeometric operators. A key result to characterize the spherical functions of of any type
is the fact that the differential operator
is conjugated, by a matrix polynomial function
, to a hypergeometric operator
. From [RT], (or [PT2], for a more general situation) we obtain that the function
, where
![∑ (i) ∑ i X = j Eij T = (1 - t) Eii, 0≤j≤i≤ℓ 0≤i≤ℓ](/img/revistas/ruma/v49n2/2a02237x.png)
![D = ψ -1 ˜D ψ](/img/revistas/ruma/v49n2/2a02238x.png)
![]() |
where the coefficient matrices are
This fact allows us to describe the eigenfunctions of the differential operator in term of the matrix valued hypergeometric functions, introduced in [T2]: Let
be a
-dimensional complex vector space, and let
and
. The hypergeometric equation is
where stands for a function of
with values in
.
More generally we can consider the equation
![]() |
In the scalar case the differential operator (11) is always of the form (10). Nevertheless in a noncommutative setting the equations and
may have no solutions
,
.
If the eigenvalues of are not in
we define the function
![]() |
where the symbol is defined inductively by
and
![]() |
for all . The function
is analytic on
, with values in
. Moreover if
then
is a solution of the hypergeometric equation (11) such that
. Conversely any solution
of (11), analytic at
is of this form.
2.2. Spherical functions as matrix hypergeometric functions. The irreducible spherical functions of of type
are in a one to one correspondence with certain simultaneous
-polynomial eigenfunctions
of the differential operators
and
(see Theorem 2.1).
A delicate fact establish in [RT] is that the functions are also polynomials functions which are eigenfunctions of the differential operators
and
.
In the variable , these operators have the form
![]() |
![]() |
where the coefficient matrices are
![]() | (15) |
To describe all simultaneous -polynomial eigenfunctions of the differential operators
and
we start by considering the eigenfunctions of
of eigenvalues
, with
(see (8)). We let
![Vλ = {F = F (u) : DF = λF, F polynomial}.](/img/revistas/ruma/v49n2/2a02290x.png)
Remark. It is not difficult to prove that that if and only if
, for some
.
Therefore if then it is of the form
![(U ;V+ λ ) F(u ) = 2H1 C ;u F0,](/img/revistas/ruma/v49n2/2a02295x.png)
![F0 ∈ ℂ ℓ](/img/revistas/ruma/v49n2/2a02296x.png)
![D](/img/revistas/ruma/v49n2/2a02297x.png)
![E](/img/revistas/ruma/v49n2/2a02298x.png)
![F0](/img/revistas/ruma/v49n2/2a02299x.png)
Since the initial value determines
, we have that the linear map
defined by
is a surjective isomorphism. Since
and
commute, the differential operators
and
also commute. Moreover, since
has polynomial coefficients whose degrees are less or equal to the corresponding orders of differentiation,
restricts to a linear operator of
. Thus we have the following commutative diagram
![]() |
where is the
matrix given by
![]() |
The eigenvalues of are given by (see Theorem 10.3 in [GPT1])
![μk (λ ) = λ(n - ℓ + 3k) - 3k(ℓ - k + 1)(n + k + 1), k = 0,1, ...,ℓ.](/img/revistas/ruma/v49n2/2a02316x.png)
![μk (λ)](/img/revistas/ruma/v49n2/2a02317x.png)
![M (λ)](/img/revistas/ruma/v49n2/2a02318x.png)
![v = (v0,...,vℓ)t](/img/revistas/ruma/v49n2/2a02319x.png)
![μ](/img/revistas/ruma/v49n2/2a02320x.png)
![M (λ )](/img/revistas/ruma/v49n2/2a02321x.png)
![v0 ⁄= 0](/img/revistas/ruma/v49n2/2a02322x.png)
The irreducible spherical functions of of type
are parameterized by two nonnegative integers
with
and
(see (7)). Under this correspondence the function
associated to the spherical function satisfies
and
where
![]() |
Then the characterization of the irreducible spherical functions is summarize in the following theorem, taking from [RT].
Theorem 2.2. The function associated to a spherical function of type
and parameters
is of the form
, where
![]() |
and is the unique
-eigenvector of
normalized by
. The expressions of the matrices
are given in (15) and the eigenvalues
and
are given in (18).
2.3. Orthogonality. Let be the space of all continuous functions
such that
for all
,
. Let us equip
with an inner product such that
becomes unitary for all
. We have the following inner product in
:
![]() |
where denotes the adjoint of
with respect to the inner product in
. Then we have the following inner product on the corresponding functions
's associated to the spherical functions
![]() |
where
![˜ ∑ n+ℓ-i W (t) = (1 - t)t Eii. 0≤i≤ℓ](/img/revistas/ruma/v49n2/2a02360x.png)
Since the Casimir operator is symmetric with respect to the -inner product for matrix valued functions on
given in (20), it follows that the differential operators
and
are symmetric with respect to the weight function
, that is they satisfy
![⟨DH, K ⟩ = ⟨H, DK ⟩.](/img/revistas/ruma/v49n2/2a02366x.png)
Now it is easy to verify that the differential operators and
are symmetric with respect to the weight function
![]() |
To illustrate the above result we will display the cases (the scalar case) and
, where the size of our matrices will be
.
3.1. The case .. In this case the functions
are scalar functions. If the parameter
is 0 then we have the zonal spherical functions.
The operator is proportional to
, (
) and
![D˜h = t(1 - t)h′′ + (n + 1 - t(n + 3 ))h ′](/img/revistas/ruma/v49n2/2a02380x.png)
![˜D](/img/revistas/ruma/v49n2/2a02381x.png)
![λ = - w (w + n + 2)](/img/revistas/ruma/v49n2/2a02382x.png)
![h](/img/revistas/ruma/v49n2/2a02383x.png)
![a = - w, b = w + n + 2 and c = n + 1.](/img/revistas/ruma/v49n2/2a02384x.png)
![]() |
are linearly independent solutions. By Theorem 2.1 we have that should be a polynomial function such that
. Moreover if
the function
have to satisfies
with
a polynomial function. Therefore we get: For
and
![(n+1)w- (- w,w+n+2 ) hw (t) = (-w -1)w 2F1 n+1 ;t](/img/revistas/ruma/v49n2/2a02394x.png)
![n < 0](/img/revistas/ruma/v49n2/2a02395x.png)
![w = - n,- n + 1,...](/img/revistas/ruma/v49n2/2a02396x.png)
![( ) hw(t) = --(1-n)w+n--t-n2F1 -w -1n-,wn+2 ;t (-w-n- 1)w+n](/img/revistas/ruma/v49n2/2a02397x.png)
![(a)m = a(a + 1) ...(a + m - 1)](/img/revistas/ruma/v49n2/2a02398x.png)
![m ∈ ℕ](/img/revistas/ruma/v49n2/2a02399x.png)
![(a)0 = 1](/img/revistas/ruma/v49n2/2a02400x.png)
By using the Pfaff's identity we get
![(n+1)w ( -w,w+n+2 ) (-w,w+n+2 ) (-w-1)w2F1 n+1 ;t = 2F1 2 ;1 - t](/img/revistas/ruma/v49n2/2a02401x.png)
![-(1-n)w+n----n ( -w- n,w+2 ) - n (- w-n,w+2 ) (- w-n-1)w+nt 2F1 1- n ;t = t 2F1 2 ;1 - t](/img/revistas/ruma/v49n2/2a02402x.png)
![( ) ( ) t-n2F1 -w-n,w+2;1 - t = 2F1 -w,w+n+2 ;1 - t 2 2](/img/revistas/ruma/v49n2/2a02403x.png)
Therefore we obtain that
Proposition 3.1. The spherical functions associated to the complex projective plane of type are
![( -w,w+n+2 ) hw (t) = 2F1 2 ;1 - t](/img/revistas/ruma/v49n2/2a02405x.png)
![w ∈ ℤ](/img/revistas/ruma/v49n2/2a02406x.png)
![w ≥ 0](/img/revistas/ruma/v49n2/2a02407x.png)
![w + n ≥ 0](/img/revistas/ruma/v49n2/2a02408x.png)
![˜ ˜ Dhw = - w (w + n + 2 )hw Ehw = - nw (w + n + 2)hw.](/img/revistas/ruma/v49n2/2a02409x.png)
3.2. The case . In this case the operators
and
are
In [GPT1], Section 11.1 we exhibit the complete list of spherical function of type . We have two families of such functions, corresponding with the choice of the parameter
or
. For
the parameter
is in the range
, and if
.
First family. For we have
,
. The (vector valued) function
is given by, up to the normalizing constant such that
.
![( (( --λ-) ( -w, w+n+3, λ-n ) ( -w, w+n+3 ) ) ||{ 1 - n+1 3F2 n+2, λ-n- 1 ;t , 2F1 n+1 ;t if n ≥ 0 H = || ( ( w+2, -w-n- 1, a+1 ) ( w+3, -w -n ) ) ( nt- n-13F2 -n, a ;t , t-n 2F1 1-n ;t if n < 0](/img/revistas/ruma/v49n2/2a02427x.png)
with .
Second family. For we have
and
. The functions
is
![( ( ( ) ( )) || 2F1 -w,nw++2n+4 ;t , - (n + 1) 3F2 - w-n1+,1 w,+nλ+-31, λ;t if n ≥ 0 { H = | ( ) |( t-n- 1F (w+3, -w- n-1;t) ,-bt- n F (w+3, - w-n-1, b+1;t) if n < 0 2 1 -n n 3 2 1-n, b](/img/revistas/ruma/v49n2/2a02433x.png)
with .
By taking the Taylor expansion at these functions takes the following unified expression. We recall that
corresponds to the identity of the group
.
Theorem 3.2. The complete list of spherical functions associated to of type
are given by
- For
we have
,
and
The parameter
is an integer that satisfies
and
.
- For
, we have
,
and
where
.
The parameteris an integer that satisfies
and
.
In this case the function , where
and
is
![]() |
In the variable , the conjugated operators
and
are
![]() |
The matrix (see (17)), is
![( 1 9 ) M (λ) = λλ(n + 2) 1 2 . λ(-2 - n - 2) λ (n + 2) - 3(n + 2)](/img/revistas/ruma/v49n2/2a02464x.png)
![M (λ )](/img/revistas/ruma/v49n2/2a02465x.png)
![μ0 = λ(n - 1)](/img/revistas/ruma/v49n2/2a02466x.png)
![μ1 = (n + 2)(λ - 3)](/img/revistas/ruma/v49n2/2a02467x.png)
![( ) ( ) F0,0 = 1λ , F0,1 = λ-21(n+2) . - 3 ---3----](/img/revistas/ruma/v49n2/2a02468x.png)
![F](/img/revistas/ruma/v49n2/2a02469x.png)
![k = 0,1](/img/revistas/ruma/v49n2/2a02470x.png)
![( ) F (u) = 2H1 U ;V +λk;u F0,k, C](/img/revistas/ruma/v49n2/2a02471x.png)
![λk = - w(w + n + 3 + k) - k(n + k + 1)](/img/revistas/ruma/v49n2/2a02472x.png)
![( ) ( ) ( ) C = 2 0 , U = n + 4 0 , V = 0 1 . 1 4 0 n + 5 0 - n - 2](/img/revistas/ruma/v49n2/2a02473x.png)
The explicit expression of the entries of these functions 's are given in the following theorem.
Theorem 3.3. The functions associated to the spherical functions of the pair
of type
are given by
- For
we have
,
and
The parameter
is an integer that satisfies
and
.
- For
, we have
,
and
where
.
The parameteris an integer that satisfies
and
.
Proof. If is an eigenfunction of
then
is an eigenfunction of
with the same eigenvalue. Explicitly the function
is
![]() |
From
![]() |
(26) we only have to prove the expression for the second entry of the function . For the first family, from Theorem 3.2 we get
For the second family we obtain, with
This concludes the proof of the theorem. □
3.3. Matrix valued orthogonal polynomials coming from spherical functions. In the scalar case, it is well known that the zonal spherical functions of the sphere are given, in spherical coordinates, in terms of Gegenbauer polynomials. Therefore, it is not surprising that in the matrix valued setting the same phenomenon occurs: the matrix spherical functions are closely related to matrix orthogonal polynomials.
For a given nonnegative integers and
we define the matrix polynomial
as the
matrix function whose
-row is the polynomial
, associated to the spherical functions of type
, given in the previous section. In other words
![( ( ) ( )) 3F2 -w,w+3n,+13, 2;u w(w+n+3-)2F1 -w+1,w4+n+4 ;u | 3 | Pw (u) = |( ( -w,w+n+4 ) sw (- w,w+n+4,sw+1 ) |) . 2F1 3 ;u - 3 3F2 4,sw ;u](/img/revistas/ruma/v49n2/2a02511x.png)
![s = w (w + n + 4) + 3(n + 1) w](/img/revistas/ruma/v49n2/2a02512x.png)
Since different spherical functions are orthogonal with respect to the natural inner product among these functions, we obtain that the matrices are orthogonal with respect to the weight function
:
![( ) W (u) = u(1 - u)n 2 - u u , u u2](/img/revistas/ruma/v49n2/2a02515x.png)
![∫ 1 * ′ (Pw, Pw′) = Pw(u)W (u)Pw′(u) du = 0, for all w ⁄= w . 0](/img/revistas/ruma/v49n2/2a02516x.png)
![Pw](/img/revistas/ruma/v49n2/2a02517x.png)
![(Pw)w](/img/revistas/ruma/v49n2/2a02518x.png)
![W](/img/revistas/ruma/v49n2/2a02519x.png)
The columns of are eigenfunctions of the differential operators
and
given in (25), thus we have that
satisfies
![( ) ( ) DPw (u )* = Pw (u)* λ0(0w)λ 0(w ) , EPw (u )* = Pw (u)* μ0(0w)μ 0(w ) , 1 1](/img/revistas/ruma/v49n2/2a02524x.png)
![λk(w )](/img/revistas/ruma/v49n2/2a02525x.png)
![μk (w)](/img/revistas/ruma/v49n2/2a02526x.png)
![pict](/img/revistas/ruma/v49n2/2a02527x.png)
3.4. Extension of the group parameters. These results have a direct and fruitful generalization by replacing the complex projective plane by the -dimensional complex projective space
, which can be realized as the homogeneous space
, where
and
.
In this case, the finite dimensional irreducible representations of , are parameterized by the
-tuples of integers
such that
. By considering the irreducible spherical functions of type
and proceeding as we explained for the complex projective plane, one obtains a situation that generalizes the one of
. Then by extending the parameters
,
we have the following results.
Theorem 3.4. Let and let us define
![( ( ) ( )) 3F2 -w,w+βα++2,1β+2,2;1 - t w(w+αβ++2β+2)2F1 -w+1,wβ++α3+ β+3;1 - t Pw (t) = |( F ( -w,w+ α+β+3 ;1 - t) - sw--F ( -w,w+α+ β+3,sw+1;1 - t) |) 2 1 β+2 β+2 3 2 β+3,sw](/img/revistas/ruma/v49n2/2a02542x.png)
Then
![{Pw }w≥0](/img/revistas/ruma/v49n2/2a02543x.png)
![( ) α β β + t β(1 - t) W (t) = t (1 - t) β(1 - t) β(1 - t)2 , (α,β > - 1).](/img/revistas/ruma/v49n2/2a02544x.png)
Let be the following second order differential operator
![2 ( ) ( ) D = t(1 - t) d-+ α+2 -t(α+β+3) 0 d-+ 0 β I dt2 -1 α+1-t(α+ β+4) dt 0 -(α+β+1)](/img/revistas/ruma/v49n2/2a02546x.png)
![P w](/img/revistas/ruma/v49n2/2a02547x.png)
![DP *w = P *wΛw](/img/revistas/ruma/v49n2/2a02548x.png)
![( - w (w + α + β + 2) 0 ) Λw = 0 - w(w + α + β + 3) - (α + β + 1)](/img/revistas/ruma/v49n2/2a02549x.png)
In [PT1] for or in general in [P08], we obtain a multiplication formula for spherical functions by tensoring certain irreducible representations of
and decomposing them into irreducible representations. From this formula we derive a three term recursion relation for the "packages" of spherical functions. Restricting this to the variable
(the variable that parameterizes a section of the
-orbits in
), we obtain a three term recursion relation for the packages of functions
associated to the spherical functions. In this case we obtain the following
Theorem 3.5. The sequence satisfies the following three term recursion relation
![]() |
with
![( ------w-(w+-α)(w+α+-β+1)----- ----------wβ---------- ) A = (w+ α+β)(2w+α+ β+1)(2w+α+ β+2) (w+1)(w+ α+β)(2w+ α+β+2) w 0 ------w(w+2-)(w+α+1)------- (w+1 )(2w+α+β+2)(2w+ α+β+3)](/img/revistas/ruma/v49n2/2a02558x.png)
![( (w+1)(w+β+2)(w+ α+β+2) ) (w+2)(2w+-α+β+2)(2w+-α+β+3) 0 Cw = --------(w+β+2)--------- --(w+β+2)(w+-α+β+1)(w+α+β+3)-- (w+2 )(w+α+ β+2)(2w+α+ β+3) (w+α+β+2 )(2w+α+β+3)(2w+ α+β+4)](/img/revistas/ruma/v49n2/2a02559x.png)
![( ) B11 -------β(w+α+β+2-)------ Bw = ---------w+wα+1---------- (w+2)(w+α+β+12)2(2w+ α+β+2) (w+1)(w+α+β+1 )(2w+α+β+3) B w](/img/revistas/ruma/v49n2/2a02560x.png)
![pict](/img/revistas/ruma/v49n2/2a02561x.png)
Remark 3.6. The three term recursion relation can be seen as a difference operator in the variable , given by a semiinfinite matrix
. The vector matrix
is an eigenfunction of
because it satisfies
.
We observe that the semiinfinte matrix have the interesting property that the sum of all the matrix elements in any row is equal to one. Moreover all the entries of
are nonnegative real numbers. This have important applications in the modeling of some stochastic phenomena.
[GV] Gangolli R. and Varadarajan V. S. Harmonic analysis of spherical functions on real reductive groups, Springer-Verlag, Berlin, New York, 1988. Series title: Ergebnisse der Mathematik und ihrer Grenzgebiete, 101. [ Links ]
[GPT1] F. A. Grünbaum, I. Pacharoni and J. Tirao, Matrix valued spherical functions associated to the complex projective plane, J. Funct. Anal. 188 (2002), 350-441. [ Links ]
[P08] Pacharoni I.Three term recursion relation for spherical functions. Preprint, 2008. [ Links ]
[PT1] Pacharoni I. and Tirao J. A. Three term recursion relation for spherical functions associated to the complex projective plane. Math Phys. Anal. Geom. 7 (2004), 193-221. [ Links ]
[PT2] Pacharoni I. and Tirao J. A. Matrix valued orthogonal polynomials arising from the complex projective space. Constr. Approxim. 25, No. 2 (2007) 177-192. [ Links ]
[PR] Pacharoni, I. Román, P. A sequence of matrix valued orthogonal polynomials associated to spherical functions Constr. Approxim. 28, No. 2 (2008) 127-147. [ Links ]
[RT] P. Román and J. A. Tirao. Spherical functions, the complex hyperbolic plane and the hypergeometric operator. Intern. J. Math. 17, No. 10, (2006), 1151-1173. [ Links ]
[T1] J. Tirao. Spherical Functions. Rev. de la Unión Matem. Argentina, 28 (1977), 75-98. [ Links ]
[T2] J. Tirao, The matrix-valued hypergeometric equation. Proc. Natl. Acad. Sci. U.S.A., 100 No. 14 (2003), 8138-8141. [ Links ]
I. Pacharoni
CIEM-FaMAF,
Universidad Nacional de Córdoba,
Córdoba 5000, Argentina
pacharon@mate.uncor.edu
Recibido: 18 de mayo de 2008
Aceptado: 11 de agosto de 2008