PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 133, Number 7, Pages 1965–1976
S 0002-9939(05)07885-8
Article electronically published on February 24, 2005
RODRIGUES TYPE FORMULA
FOR ORTHOGONAL POLYNOMIALS ON THE UNIT BALL
YUAN XU
(Communicated by Jonathan M. Borwein)
Abstract. For a class of weight functions invariant under reflection groups
on the unit ball, a family of orthogonal polynomials is defined via a Rodrigues
type formula using the Dunkl operators. Their properties and their relation
with several other bases are explored.
1. Introduction
Let Πd = R[x1 , . . . , xd ] denote the space of polynomials in d variables and Πdn
denote the subspace of polynomials of degree at most n. We will use the standard
αd
1
multi-index notation. For α ∈ Nd0 and x ∈ Rd , xα = xα
1 · · · xd is a monomial of
degree |α| = α1 + . . . + αd .
The classical orthogonal polynomials on the unit ball B d = {x : x ≤ 1}, where
x is the Euclidean norm of x, are orthogonal with respect to the weight function
(1.1)
Wµ (x) = (1 − x2 )µ−1/2 ,
x ∈ Bd,
µ > −1/2.
The study of these polynomials can be traced back to Hermite; see [1] and [5,
Chapt. 12]. Among the first orthogonal bases on B d being studied is the one given
by the Rodrigues type formula
(1.2)
Uα (x) = (1 − x2 )−µ+1/2 ∂ α (1 − x2 )n+µ−1/2 ,
α ∈ Nd0 ,
where ∂i = ∂/∂xi and ∂ α = ∂1α1 · · · ∂dαd , and a family of polynomials that are
biorthogonal to Uα .
The purpose of this paper is to consider the analogues of Uα for orthogonal
polynomials on B d with respect to a class of weight functions invariant under a
reflection group. We first define the weight function. Let G be a finite reflection group with a fixed positive root system R+ . Let σv denote the reflection
along v ∈ R+ , that is, xσv = x − 2x, v/v2 for x ∈ Rd with the inner product
x, y = di=1 xi yi . Thus, G is a finite orthogonal group generated by the reflections
{σv : v ∈ R+ }. Let κ be a multiplicity function defined on R+ , that is, κ : R+ → R
Received by the editors February 13, 2003.
2000 Mathematics Subject Classification. Primary 33C50, 42C10.
Key words and phrases. Orthogonal polynomials on a unit ball, Rodrigues type formula,
biorthogonal polynomials, reflection invariant weight function.
This work was partially supported by the National Science Foundation under Grant DMS0201669.
c
2005
American Mathematical Society
1965
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1966
YUAN XU
is a G-invariant function. We assume that κ(v) ≥ 0 for all v ∈ R+ . Then the
function
(1.3)
hκ (x) =
|x, v|κ(v) ,
x ∈ Rd ,
v∈R+
is a positive homogeneous G-invariant function of order γ := γκ =
v∈R+ κv .
For example, if G is the symmetric group, then hκ (x) = 1≤i<j≤d |xi − xj |κ and
γ = d(d − 1)κ/2. In this paper we consider the reflection invariant weight function
Wκ,µ on the unit ball
(1.4)
Wκ,µ (x) = h2κ (x)(1 − x2 )µ−1/2 ,
x ∈ Bd,
µ > −1/2
where h2κ is as in (1.3). If κ = 0, then Wκ,µ (x) becomes Wµ (x) in (1.1).
The weight function hκ was introduced by Dunkl for the purpose of studying the
orthogonal structure for polynomials on the unit sphere S d−1 = {x ∈ Rd : x = 1}
with respect to the measure h2κ (x)dω, where dω is the rotation invariant measure
on S d−1 . The main ingredient in the study is a family of commuting differentialdifference operators, Di , called Dunkl’s operators ([2]),
f (x) − f (xσv )
Di f (x) = ∂i f (x) +
ei , v,
κv
1 ≤ i ≤ d,
x, v
v∈R+
where ei = (0, . . . , 0, 1, 0, . . . , 0) with 1 in the i-th component. The set {Di : 1 ≤
i ≤ d} generates a commutative algebra of operators containing the h-Laplacian
∆h = D12 +. . .+Dd2 . Let Pnd denote the space of homogeneous polynomials of degree
n in d variables. If p ∈ Pnd , then
p(x)q(x)h2 (x)dω = 0,
∀q ∈ Πdn−1 ,
S d−1
if and only if ∆h p = 0. The space Hnd (h2κ ) := Pnd ∩ ker ∆h is called the space of hharmonic polynomials of degree n. If κ(v) = 0, it agrees with the space of ordinary
harmonic polynomials Hnd := Pnd ∩ ker ∆, where ∆ denote the ordinary Laplacian
∆ = ∂12 + . . . + ∂d2 . For the extensive study of h-harmonics and the related work,
see [4] and the references therein.
The theory of h-harmonics has also made it possible to study the orthogonal
structure of polynomials with respect to Wκ,µ on B d . Initial work in this direction
has been carried out in [12, 13], which has led to new results even in the case of
the classical weight function Wµ . In the present paper we show that the orthogonal
polynomials Uα (x) in (1.2) defined by the Rodrigues type formula can be extended
to the reflection invariant weight function Wκ,µ . That is, we define Uα as follows:
Definition 1.1. For α ∈ Nd0 , define
Uα (x) = (1 − x2 )−µ+1/2 Dα (1 − x2 )|α|+µ−1/2 .
We will show that Uα (x) so defined are orthogonal polynomials with respect to Wκ,µ
on B d . Properties of these polynomials and their relations with other orthogonal
bases will be discussed. Let us mention that recently Dunkl and Dunkl-Cherednik
operators have been used to give the Rodrigues type formula for nonsymmetric Hermite and Laguerre polynomials with respect to reflection invariant weight functions
([8, 10]) and for Macdonald’s polynomials ([7]).
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ORTHOGONAL POLYNOMIALS ON THE UNIT BALL
1967
2. Rodrigues type formula and orthogonal bases
For n ∈ Nd0 let Vnd (Wκ,µ ) denote the subspace of polynomials of degree exactly
n in Πd that are orthogonal with respect to Wκ,µ . A polynomial p ∈ Vnd (Wκ,µ ) if
p ∈ Πdn and B d p(x)q(x)Wκ,µ (x)dx = 0 for all q ∈ Πdn−1 . We show Uα ∈ Vnd (Wκ,µ ).
The definition of Uα in Definition 1.1 shows that it depends on κ and µ. In the
following proposition, we write Uαµ := Uα to emphasize its dependence on µ.
Proposition 2.1. Let α ∈ Nd0 and |α| = n. Then Uαµ is a polynomial of degree n,
and we have a recursive formula
µ
(x) = −(2µ + 1)xi Uαµ+1 (x) + (1 − x2 )Di Uαµ+1 (x).
Uα+e
i
Proof. Since (1 − x2 )a is invariant under the reflection group, a simple computation shows that
Di Uαµ+1 (x) = ∂i (1 − x2 )−µ−1/2 Dα (1 − x2 )|α|+µ+1/2
+ (1 − x2 )−µ−1/2 Dα+ei (1 − x2 )|α|+µ+1/2
µ
= (2µ + 1)xi (1 − x2 )−1 Uαµ+1 (x) + (1 − x2 )−1 Uα+e
(x),
i
which proves the recursive relation. That Uαµ is a polynomial of degree |α| is a
consequence of this relation (use induction if necessary).
The following proposition plays a key role in proving that Uα is in fact an orthogonal polynomial with respect to Wκ,µ on B d .
Proposition 2.2. Assume that p and q are continuous functions and p vanishes
on the boundary of B d . Then
2
Di p(x)q(x)hκ (x)dx = −
p(x)Di q(x)h2κ (x)dx.
Bd
Bd
Proof. The proof is similar to the proof of Lemma 3.7 of [3] (Theorem 5.1.18 of [4]).
We shall be brief. Assume κv ≥ 1. Analytic continuation can be used to extend
the range of validity to κv ≥ 0. Integration by parts shows
∂i p(x)q(x)h2κ (x)dx = −
p(x)∂i (q(x)h2κ (x))dx
d
d
B
B
2
=−
p(x)∂i q(x)hκ (x)dx − 2
p(x)q(x)hκ (x)∂i hκ (x)dx.
Bd
Bd
Let Di denote the difference part of Di , Di = ∂i + Di . For a fixed root v, a simple
computation shows that
q(x) + q(xσv ) 2
hκ (x)dx.
Di p(x)q(x)h2κ (x)dx =
p(x)
κi vi
x, v
d
d
B
B
v∈R+
Adding these two equations and using the fact that
1
hκ (x)∂i hκ (x) =
h2 (x)
κv vi
v, x κ
v∈R+
completes the proof.
d
Theorem 2.3. For α ∈ Nd0 , the polynomials Uα are elements in V|α|
(Wκ,µ ).
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1968
YUAN XU
Proof. Let n = |α|. For each β ∈ Nd0 and |β| < n, we claim that
(2.1)
Dβ (1 − x2 )n+µ−1/2 = (1 − x2 )n−|β|+µ−1/2 Qβ (x)
for some Qβ ∈ Πdn . This follows from induction. The case β = 0 is trivial with
Q0 (x) = 1. Suppose the equation is true for |β| < n − 1. Then
Dβ+ei (1 − x2 )n+µ−1/2 = Di (1 − x2 )n−|β|+µ−1/2 Qβ (x)
= ∂i (1 − x2 )n−|β|+µ−1/2 Qβ (x) + (1 − x2 )n−|β|+µ−1/2 Di Qβ (x)
= (1 − x2 )n−|β|+µ−3/2 −2(n − |β| + µ − 1/2)xi Qβ (x) + (1 − x2 )Di Qβ (x)
= (1 − x2 )n−|β+ei |+µ−1/2 Qβ+ei (x),
where Qβ+ei , as defined above, is clearly a polynomial of degree |β| + 1, which
completes the inductive proof. This formula shows, in particular, that the function
Dβ (1−x2 )n+µ−1/2 vanishes on the boundary of B d if |β| < n. For any polynomial
p ∈ Πdn−1 , using Proposition 2.2 repeatedly gives
Uα (x)p(x)h2κ (x)(1 − x2 )µ−1/2 dx =
Dα (1 − x2 )n+µ−1/2 p(x)h2κ (x)dx
Bd
Bd
= (−1)n
Dα p(x)(1 − x2 )n+µ−1/2 h2κ (x)dx = 0,
Bd
d
since Di : Pnd → Pn−1
, which implies Dα p(x) = 0 as p ∈ Πdn−1 .
By definition, Uα is orthogonal to polynomials of lower degree. However, if
|α| = |β|, Uα and Uβ are not necessarily orthogonal. In fact, they are not and there
is another family of orthogonal polynomials in Vnd (Wκ,µ ) that are biorthogonal to
d
d
Uα . For its definition, we need the projection operator projB
n : Pn → Vn (Wκ,µ ).
d
This operator has an explicit formula ([12]). For p ∈ Pn ,
1
∆j p(x),
(2.2)
projB d p(x) =
4j j!(−n − λ − µ + 1)j h
0≤j≤n/2
where λ = γκ +(d−1)/2 and (a)j = a(a+1) . . . (a+j−1) is the Pochhammer symbol.
Let Vκ be the intertwining operator between the family of partial derivatives and
Dunkl’s operators, which is a linear operator uniquely determined by ([3])
Vκ 1 = 1,
Vκ Pnd = Pnd ,
Di Vκ = Vκ ∂i ,
1 ≤ i ≤ d.
The explicit formula of Vκ , however, is known only in the case of the group Zd2 and
S3 , the symmetric group of three variables. For each β ∈ Nd0 , the function Vκ xβ is
a homogeneous polynomial of degree |β|. Define


1
β

Rβ (x) := projB
∆j xβ  ,
n Vκ {x } = Vκ
4j j!(−n − λ − µ + 1)j
0≤j≤n/2
where the second equality follows from (2.2) and the fact that ∆h Vκ = Vκ ∆. Clearly
Rβ is an orthogonal polynomial of degree n with respect to Wκ,µ and {Rβ : |β| = n}
is a basis of Vnd (Wκ,µ ). It turns out that Rβ and Uα are biorthogonal. For α ∈ Nd0 ,
write α! = α1 ! . . . αd !.
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ORTHOGONAL POLYNOMIALS ON THE UNIT BALL
1969
Theorem 2.4. The two families {Uα : |α| = n, α ∈ Nd0 } and {Rα : |α| = n, α ∈ Nd0 }
are biorthogonal with respect to Wκ,µ on B d ; more precisely,
(µ + 1/2)n
aκ
Uα (x)Rβ (x)Wκ,µ (x)dx = Aα δα,β ,
Aα = (−1)n α!
.
(λ
+ µ + 1/2)n
d
B
In particular, {Uα : |α| = n} is a basis for Vnd (Wκ,µ ).
Proof. Since both Uα and Vβ are orthogonal polynomials with respect to Wκ,µ , we
only need to consider the case that they are both polynomials of degree n. Let
α, β ∈ Nd0 and |α| = |β|. Following the proof of Proposition 2.3, we have
Uα (x)Rβ (x)h2κ (x)(1 − x2 )µ−1/2 dx
aκ
Bd
Dα [Rβ (x)] (1 − x2 )n+µ−1/2 h2κ (x)
= (−1)n aκ
d
B
n
= (−1) aκ
Dα Vκ xβ (1 − x2 )n+µ−1/2 h2κ (x)
Bd
n
= δα,β α!(−1) aκ
(1 − x2 )n+µ−1/2 h2κ (x),
Bd
since Rβ (x) = x + lower degree polynomials and Dα Vκ xβ = Vκ ∂ α xβ = α!δα,β Vκ 1
= α!δα,β . The constant is computed by
1
(1 − x2 )n+µ−1/2 h2κ (x) =
rd−1+2γ (1 − r2 )µ−1/2 dr
h2κ (x )dω(x )
0
Bd
S d−1
Γ(γ + d/2)Γ(µ + 1/2)
=
h2 (x )dω(x )
2Γ(γ + µ + (d + 1)/2) S d−1 κ
(µ + 1/2)n
a−1 ,
=
(λ + µ + 1/2)n κ
β
since a−1
κ is just the above integral with n = 0. Finally, it follows from the biorthogonality that the set {Uα : |α| = n} is linearly independent, so that it is a basis of
Vnd (Wκ,µ ).
For our next property of Uα (x), we will need the reproducing kernel Pn (x, y) of
Vnd (Wκ,µ ). This kernel is characterized by the property that
aκ
P (x, y)q(y)Wκ,µ (y)dy = q(x),
∀q ∈ Vnd (Wκ,µ ).
Bd
It is shown in [12] that the kernel enjoys a closed formula given by
n+λ+µ
cµ
(2.3) Pn (x, y) =
λ+µ
1
Cnλ+µ (x, · + t 1 − x2 1 − y2 )(1 − t2 )µ−1 dt (y),
× Vκ
1
−1
where cµ = 1/ −1 (1 − t2 )µ−1 dt and Vκ acts on the variable represented by the dot
in the formula. This closed formula for the reproducing kernel has been used to
study summability of the Fourier orthogonal expansion on the unit ball. It follows
from the biorthogonal property of {Uα } and {Rβ } that
(2.4)
Pn (x, y) =
A−1
α Rα (x)Uα (y).
|α|=n
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1970
YUAN XU
Indeed, multiplying the right-hand side of the above equation by Rβ (y) and integrating, the result is Rβ (y) by the biorthogonality, which shows that the right-hand
side satisfies the reproducing property. Since the kernel is unique, it has to equal
the left-hand side.
Let Kn (x, y) = Vκ (x, yn )/n!. It is known ([3]) that
Kn (x, D(y) )p(y) = p(x),
(2.5)
∀p ∈ Pnd ,
where D(y) means that Di is taken with respect to the variable y and Kn (x, D(y) )
(y)
is the operator formed by replacing yi by Di in Kn (x, y). A bilinear form p, qh
is defined on Pnd by
p, qh = p(D)q(x),
p, q ∈ Pnd .
This form is symmetric in the sense that p, qh = q, ph ([3]). We have the
following lemma that will be useful below.
Lemma 2.5. Let α ∈ Nd0 with |α| = n and q ∈ Pjd with j < n. Then
(D(y) )α q(y)Vκ(y) x, yn−j = (n − j)!q(D)xα .
Proof. Let β ∈ Nd0 and |β| = j. As a function in y, y β Vκ x, yn−j is a polynomial
in Pnd . Using the fact that the bilinear form is symmetric, we get
(D(y) )α y β Vκ x, yn−j /(n − j)! = y α , y β Kn−|β| (x, y)h
= y β Kn−|β| (x, y), y α h = Kn−|β|(x, D(y) )Dβ y α = Dβ xα ,
where the last step follows from the equation (2.5) with p(y) = Dβ y α and n replaced
by n − j.
Proposition 2.6. Let α ∈ Nd0 and |α| = n. Then
Uα (x) =
0≤2j≤n
2n (µ + 1/2)n
(−1)n−j (1 − x2 )j ∆jh xα .
22j (µ + 1/2)j j!
In particular, Uα (x) = (−1)n 2n (µ + 1/2)n xα + (1 − x2 )Q(x), where Q ∈ Πdn−2 .
Proof. Since Dα Rβ (y) = α!δα,β , applying (D(y) )α to the equation (2.4) gives
(y) α
α!A−1
) Pn (x, y).
α Uα (x) = (D
We then use the expression of Pn (x, y) in (2.3). Since the leading coefficient of
Cnλ (t) is (λ)n 2n /n!, it follows that Pn (x, y), as a function of y, satisfies
n + λ + µ (λ + µ)n 2n
cµ
λ+µ
n!
1
n
2 µ−1
2
2
× Vκ
(x, · + t 1 − x 1 − y ) (1 − t )
dt (y) + . . .
−1
n
(λ + µ + 1)n 2n
cµ
=
bj (1 − x2 )j Vκ (1 − y2 )j x, ·n−2j (y)
2j
n!
Pn (x, y) =
0≤2j≤n
+ polynomial of lower degree in y,
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ORTHOGONAL POLYNOMIALS ON THE UNIT BALL
1971
1
where we have used the binomial formula and the fact that −1 tj (1 − t2 )µ−1 dt = 0
if j is odd, and bj is given by
1
(1/2)j
b j = cµ
t2j (1 − t2 )µ−1 dt =
.
(µ + 1/2)j
−1
Consequently, since Dα p(x) = 0 for any p ∈ Πdm , m < |α|, we have
n (1/2)j
(λ + µ + 1)n 2n
U
(x)
=
(1 − x2 )j
α!A−1
c
α
µ
α
2j (µ + 1/2)j
n!
0≤j≤n
× (D(y) )α (1 − y2 )j Vκ x, ·n−2j (y) .
Using Lemma 2.5 with q(y) = y2j , we have
(D(y) )α (1 − y2 )j Vκ x, ·n−2j (y) = (−1)j (D(y) )α y2j Vκ x, ·n−2j (y)
= (−1)j (n − 2j)!∆jh xα .
Putting these equations together we conclude
Uα (x) = Aα
(λ + µ + 1)n 2n
n!(1/2)j
cµ
(−1)j
(1 − x2 )j ∆jh xα .
α!n!
(2j)!(µ + 1/2)j
0≤j≤n
Using the identity (2j)! = 22j (1/2)j j! to simplify the constants completes the proof.
We note that the expansion is still implicit in a way, since the computation of
∆jh xα can be difficult. In the case of classical orthogonal polynomials on B d , ∆h
becomes ∆, and the multinomial formula gives
j!
j! α!
∆j xα = (∂12 + . . . + ∂d2 )j xα =
∂ 2β xα =
xα−2β ,
β!
β! (α − 2β)!
|β|=j
|β|=j
from which the explicit formula of Uα (x) for Wµ follows. Since Dunkl’s operators
commute, the multinomial formula can also be used to expand ∆h xα . However,
we do not have a nice formula for Dβ xα . This is a polynomial of degree |α| − |β|
in x. For |α| = |β| it is equal to xα , xβ h and is easily seen to be a polynomial
in variable κ. However, we do not know an explicit formula for this quantity. We
mention the following recursive formula, which can be used to compute Dβ xα of
lower degree.
Proposition 2.7. Let α, β ∈ Nd0 , |β| ≤ |α|. If αi > 0, then
Dβ xα = xi Dβ xα−ei + β1 Dβ−ei xα−ei +
κv v, ei Pβ,v (D)xα−ei ,
v∈R+
where Pβ,v (y) = (y β −(yσv )β )/y, σv is a homogeneous polynomial of degree |β|−1.
Proof. We use the product rule of Dunkl’s operators ([4, p. 157])
Di (f g)(x) = g(x)Di f (x) + f (x)∂i g(x) +
κv f (xσv )v, ei v∈R+
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g(x) − g(xσv )
.
x, v
1972
YUAN XU
Using the fact that Di Kn (x, y) = xi Kn−1 (x, y), the product rule with g(y) = y β
and f (y) = K|α|−|β|(x, y) shows that
(y) Di y β K|α|−|β|(x, y) = y β xi K|α|−|β|−1(x, y) + K|α|−|β|(x, y)βi y β−ei
y β − (yσv )β
.
+
κv K|α|−|β|(x, yσv )v, ei x, v
v∈R+
Let
w∈
(y)
φα,β (x) = (Di )α y β K|α|−|β|(x, y) . Since Kn (xw, yw)
(y)
G, applying (Di )α−ei to the above equation gives
φα,β (x) = xi φα−ei ,β (x) + βi φα−ei ,β−ei (x) +
κv v, ei = Kn (x, y) for all
cτ φα−ei ,τ (xσv ),
|τ |=|β|−1
v∈R+
where (y β − (yσv )β )/x, v = |τ |=|β|−1 cτ y τ . By Lemma 2.5, φα,β (x) = Dβ xα ,
which gives the stated formula.
3. Relation to an orthonormal basis
The biorthogonality of two bases is useful for finding the orthogonal expansions
of a function. For example, for f ∈ L2 (Wκ,µ ), the Fourier orthogonal expansion of
f in terms of the basis {Uα } is given by
∞ f (x) =
bα,n A−1
U
(x),
b
=
a
f (x)Rα (x)Wκ,µ (x)dx,
α
α,n
κ
α
Bd
n=0 |α|=n
where the equality holds in the L2 norm. That the coefficient bα,n is given as above
can be easily seen upon integrating the expansion of f (x)Rα (x) over B d . Thus,
the biorthogonality provides an easy way to compute the Fourier coefficient bα,n in
the expansion with respect to Uα (x). Note that Rα depends on the intertwining
operator Vκ . The following formula proved in [11] is useful for computing integrals
of Vκ f , which works despite the lack of an explicit formula for Vκ .
Lemma 3.1. For a continuous function f on B d ,
2
Vκ f (x)hκ (x)dω(x) = Bκ
f (x)(1 − x2 )γ−1 dx
S d−1
Bd
where the constant Bκ can be determined by setting f (x) = 1.
(µ,τ )
We use this lemma to work out the expansion in one particular case. Let Cn
be polynomials defined by the generating function
1
∞
1
2 τ −1
cµ
(1
+
t)(1
−
t
)
dt
=
Cn(µ,τ ) (x)tn .
2 )µ
(1
−
2xt
+
t
−1
n=0
(t)
These are orthogonal polynomials with respect to the weight function wµ,τ (t) =
(α,β)
(t).
|t|2τ (1−t2 )µ−1/2 on [−1, 1] and they are related to the Jacobi polynomials Pn
In particular,
(µ + τ )n (µ−1/2,τ −1/2) 2
(µ,τ )
P
(2t − 1).
(3.1)
C2n (t) =
(τ + 1/2)n n
It is known that an orthonormal basis of Vnd (Wκ,µ ) is given by
d
(µ,n−2j+λ) (x)Sβh (x) : Sβh ∈ Hn−2j
(3.2)
fj,β (x) := mj C
(h2κ ),
2j
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0 ≤ 2j ≤ n ,
ORTHOGONAL POLYNOMIALS ON THE UNIT BALL
(µ,n−2j+λ)
in which c−2
is the L2 (wλ,µ , [−1, 1]) norm of C2j
j
1973
(t) and {Sβh } is an or(µ,λ)
d
thonormal basis of Hn−2j
(h2κ ). In particular, the polynomial C2n
d
element of V2n (Wκ,µ ); hence it can be written in terms of Uα (x).
(x) is an
Proposition 3.2. The following expansion holds:
(µ,λ)
C2n
(3.3)
(x)
(µ,λ)
C2n
=
(1)
1
n!
U2β (x).
(µ + 1/2)2n 22n
β!
|β|=n
d
Proof. The fact that {Uα : |α| = 2n} is a basis of V2n
(Wκ,µ ) shows that we can
write
(µ,λ)
bα A−1
C2n (x) =
α Uα (x).
|α|=2n
Using the biorthogonality in Proposition 2.4 and the fact that Rκ (x) = Vκ xα + . . .,
we can compute the coefficient bα as follows:
(µ,λ)
C2n (x)Rα (x)h2κ (x)(1 − x2 )µ−1 dx
bα = aκ,µ
Bd
(µ,λ)
= aκ,µ
C2n (x)Vκ [xα ] h2κ (x)(1 − x2 )µ−1 dx
= bκ,µ
Bd
1
0
(µ,λ)
r2n+d−1+2γ C2n (r)(1 − r2 )µ−1/2 dr
· ch
Vκ ({·}α )(x )h2κ (x )dω(x ),
S d−1
using the polar coordinates x = rx , r > 0 and x ∈ S d−1 , where
1
r2λ (1 − r2 )µ−1/2 dr
bκ,µ = 1/
0
and
ch = 1/
S d−1
h2κ (x )dω(x ).
The first integral can be computed using the L2 norm and the leading coefficient
(µ,τ )
(µ,τ )
kn
of Cn (t) (cf. [4, p. 27]),
1
(µ,λ)
bκ,µ
r2n+d−1+2γ C2n (r)(1 − r2 )µ−1/2 dr
0
−1
= bκ,µ k (µ,τ )
0
1
2
(µ + 1/2)n (λ + µ)n
(µ,λ)
,
C2n (r) (1 − r2 )µ−1/2 dr =
(λ + µ + 1)2n
α
The integral of Vκ x can be computed using Lemma 3.1,
ch
Vκ ({·}α )(x )h2κ (x )dω = aκ
xα (1 − x2 )γ−1 dx,
S d−1
Bd
2 γ−1
where aκ = 1/ B d (1 − x )
dx, which shows that the integral is zero if one
of αi is odd. If α = 2β, then the integral over B d can be evaluated using polar
coordinates and the result is
(1/2)β
Vκ ({·}α )(x )h2κ (x )dω =
.
ch
(λ
+ 1/2)β
d−1
S
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1974
YUAN XU
Putting these equations together, we have proved that
(µ + 1/2)n (µ + λ)n (1/2)β
(µ,λ)
C2n (x) =
A−1 U2β (x).
(µ + λ + 1)2n
(λ + 1/2)β 2β
|β|=n
(µ,λ)
Using the formula of C2n
the proof.
(1) ([4, p. 27]) and simplifying the coefficients completes
Let us write the expansion explicitly. Using the multinomial formula and the
Rodrigues type formula of Uβ (x), we see that the equation (3.3) gives
(µ,λ)
C2n
(x)
(µ,λ)
C2n
(1)
=
n!
1
2 −µ+1/2
D2β (1 − x2 )2n+µ−1/2
(1
−
x
)
(µ + 1/2)2n 22n
β!
|β|=n
1
=
(1 − x2 )−µ+1/2 ∆nh (1 − x2 )2n+µ−1/2 .
(µ + 1/2)2n 22n
(µ,0)
In particular, if κ = 0 and d = 1, so that λ = 0, then C2n (x) becomes the
µ
Gegenbauer polynomial C2n
(x) and the above formula is precisely the Rodrigues
formula for the Gegenbauer polynomials. Furthermore, in polar coordinates, the
h-Laplacian can be written as
1
∂2
2λ + 1 ∂
+ 2 ∆h,0 ,
+
2
∂r
r ∂r r
where ∆h,0 is the spherical part of the operator which has h-harmonics as eigenfunctions. More precisely, if Snh ∈ Hnd (h2κ ), then
∆h =
(3.4)
∆h,0 Snh (x) = −n(n + 2λ)Snh (x).
(3.5)
(µ,λ)
This leads to the following Rodrigues type formula for C2n
Proposition 3.3. For n ≥ 0,
(µ,λ)
C2n
(r)
C2n
(1)
(µ,λ)
=
1
(1 − r2 )−µ+1/2
(µ + 1/2)2n 22n
d2
2λ d
+
2
dr
r dr
(t).
n
(1 − r2 )2n+µ−1/2 .
By the relation (3.1), this gives a Rodrigues type formula for the Jacobi polynomials, which can be rewritten as
2
n
d
(α + 1)n
2β + 1 d
(α,β)
2
2 −α
Pn
(2r − 1) =
(1 − r )
+
(1 − r2 )2n+α .
(α + 1)2n n!22n
dr2
r dr
In fact, this formula was discovered by Koornwinder [6] in the course of giving an
analytic proof for his addition formula for the Jacobi polynomials.
It is possible to derive a Rodrigues type formula for other elements in the orthonormal basis (3.2). We have the following.
h
be an h-spherical harmonic in Hnd (h2κ ). Then
Proposition 3.4. Let Sn−2j
(µ,n−2j+λ)
C2j
(x)
(µ,n−2j+λ)
C2j
(1)
=
h
(x)
Sn−2j
1
2 −µ+1/2 j
2 2j+µ−1/2 h
(1
−
x
(1
−
x
)
∆
)
S
(x)
.
n−2j
h
(µ + 1/2)2j 22j
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ORTHOGONAL POLYNOMIALS ON THE UNIT BALL
1975
h
d
Proof. Let g be a differentiable function defined on [0, 1] and Sm
∈ Hm
(h2κ ). We
claim that
2
d
2λ + 2m + 1 d
h
h
∆h g(x)Sm
(x) = Sm
(x)
+
g(r),
r = x.
dr2
r
dr
h
h
h
Indeed, since Sm
is a homogeneous polynomial of degree r, Sm
(x) = rm Sm
(x )
under the polar coordinates. Therefore, using (3.4) and (3.5),
2
d
2λ + 1 d
h
h
+
(x )
∆h g(x)Sm (x) =
(rm g(r)) · Sm
dr2
r dr
h
− m(m + 2λ)g(r)rm−2 Sm
(x )
2
d
2λ + 2m + 1 d
h
=
+
(x ),
g(r)
r m Sm
dr2
r
dr
where the second equality follows from a simple computation. Clearly, we can use
the formula repeatedly to get
2
j
d
2λ + 2m + 1 d
j h
h
∆h g(x)Sm (x) = Sm (x)
+
g(r).
dr2
r
dr
Applying the above formula with g(r) = (1 − r2 )2j+µ−1/2 and m = n − 2j, the
stated formula follows from Proposition 3.3.
Acknowlegement
The author thanks a referee for his/her careful review.
References
[1] P. Appell and J. K. de Fériet, Fonctions hypergéométriques et hypersphériques, Polynômes
d’Hermite, Gauthier-Villars, Paris, 1926.
[2] C. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer.
Math. Soc. 311 (1989), 167–183. MR0951883 (90k:33027)
[3] C. Dunkl, Integral kernels with reflection group invariance, Canad. J. Math. 43 (1991), 12131227. MR1145585 (93g:33012)
[4] C. F. Dunkl and Yuan Xu, Orthogonal polynomials of several variables, Cambridge Univ.
Press, 2001. MR1827871 (2002m:33001)
[5] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher transcendental functions,
McGraw-Hill, Vol 2, New York, 1953. MR0058756 (15:419i)
[6] T. Koornwinder, Jacobi polynomials, III. An analytic proof of the addition formula, SIAM
J. Math. Anal., 6 (1975), pp. 533-543. MR0447659 (56:5969)
[7] L. Lapointe and L. Vinet, Rodrigues formulas for the Macdonald polynomials, Adv. Math.
130 (1997), 261–279. MR1472319 (99e:05128)
[8] A. Nishino, H. Ujino and M. Wadati, Rodrigues formula for the nonsymmetric multivariable
Laguerre polynomial, J. Phys. Soc. Japan 68 (1999),797–802. MR1697908 (2000c:33014)
[9] G. Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publication 23, American Mathematical Society, Providence, RI, 1975. MR0372517 (51:8724)
[10] H. Ujino and M. Wadati, Rodrigues formula for the nonsymmetric multivariable Hermite
polynomial, J. Phys. Soc. Japan 68 (1999), 391–395. MR1682683 (2000g:33016)
[11] Yuan Xu, Integration of the intertwining operator for h-harmonic polynomials associated to
reflection groups, Proc. Amer. Math. Soc. 125 (1997), 2963–2973. MR1402890 (97m:33004)
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
1976
YUAN XU
[12] Yuan Xu, Orthogonal polynomials on the ball and on the simplex for weight functions with
reflection symmetries, Constr. Approx., 17 (2001), 383-412. MR1828918 (2002e:33018)
[13] Yuan Xu, Orthogonal polynomials and summability in Fourier orthogonal series on spheres
and on balls, Math. Proc. Cambridge Phil. Soc. 31 (2001), 139–155.
MR1833079
(2002d:42031)
Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222
E-mail address: [email protected]
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