Generalized Bessel function of Type D
We write down the generalized Bessel function associated with the root system of type D by means of multivariate hypergeometric series. Our hint comes from the particular case of the Brownian motion in the Weyl chamber of type D.
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irk-123456789-1489982019-02-20T01:24:27Z Generalized Bessel function of Type D Demni, N. We write down the generalized Bessel function associated with the root system of type D by means of multivariate hypergeometric series. Our hint comes from the particular case of the Brownian motion in the Weyl chamber of type D. 2008 Article Generalized Bessel function of Type D / N. Demni // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 19 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 33C20; 33C52; 60J60; 60J65 http://dspace.nbuv.gov.ua/handle/123456789/148998 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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We write down the generalized Bessel function associated with the root system of type D by means of multivariate hypergeometric series. Our hint comes from the particular case of the Brownian motion in the Weyl chamber of type D. |
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Generalized Bessel function of Type D |
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Generalized Bessel function of Type D |
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Generalized Bessel function of Type D |
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Generalized Bessel function of Type D |
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generalized bessel function of type d |
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Generalized Bessel function of Type D / N. Demni // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 19 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 4 (2008), 075, 7 pages
Generalized Bessel function of Type D?
Nizar DEMNI
SFB 701, Fakultät für Mathematik, Universität Bielefeld, Deutschland
E-mail: demni@math.uni-bielefeld.de
Received July 01, 2008, in final form October 24, 2008; Published online November 04, 2008
Original article is available at http://www.emis.de/journals/SIGMA/2008/075/
Abstract. We write down the generalized Bessel function associated with the root system of
type D by means of multivariate hypergeometric series. Our hint comes from the particular
case of the Brownian motion in the Weyl chamber of type D.
Key words: radial Dunkl processes; Brownian motions in Weyl chambers; generalized Bessel
function; multivariate hypergeometric series
2000 Mathematics Subject Classification: 33C20; 33C52; 60J60; 60J65
1 Root systems and related processes
We refer the reader to [11] for facts on root systems. Let (V, 〈·〉) be an Euclidean space of finite
dimension m ≥ 1. A reduced root system R is a finite set of non zero vectors in V such that
1) R ∩ Rα = {α,−α} for all α ∈ R,
2) σα(R) = R,
where σα is the reflection with respect to the hyperplane Hα orthogonal to α
σα(x) = x− 2
〈α, x〉
〈α, α〉
α, x ∈ V.
A simple system S is a basis of span(R) which induces a total ordering in R. A root α is positive
if it is a positive linear combination of elements of S. The set of positive roots is called a positive
subsystem and is denoted by R+. The (finite) reflection group W is the group generated by all
the reflections σα for α ∈ R. Given a root system R with positive and simple systems R+, S,
define the positive Weyl chamber C by:
C := {x ∈ V, 〈α, x〉 > 0 ∀α ∈ R+} = {x ∈ V, 〈α, x〉 > 0 ∀α ∈ S}
and C, ∂C its closure and boundary respectively. One of the most important properties is
that the convex cone C is a fundamental domain, that is, each λ ∈ V is conjugate to one and
only one µ ∈ C. Processes related to root systems have been of great interest during the last
decade and notably the so-called Dunkl process [17]. This V -valued process was deeply studied
in a sequence of papers by Gallardo and Yor [5, 6, 7, 8] and Chybiryakov [3] and its projection
on the Weyl chamber gives rise to a diffusion known as the W -invariant or radial Dunkl process.
The generator of the latter process acts on C2
c (C) as
Lku(x) =
1
2
∆u(x) +
∑
α∈R+
k(α)
〈α,∇u(x)〉
〈α, x〉
?This paper is a contribution to the Special Issue on Dunkl Operators and Related Topics. The full collection
is available at http://www.emis.de/journals/SIGMA/Dunkl operators.html
mailto:demni@math.uni-bielefeld.de
http://www.emis.de/journals/SIGMA/2008/075/
http://www.emis.de/journals/SIGMA/Dunkl_operators.html
2 N. Demni
with 〈α,∇u(x)〉 = 0 whenever 〈α, x〉 = 0, where k is a positive function defined on the orbits
of the W -action on V and is constant on each orbit. Such a function is called a multiplicity
function. The semi group density (with respect to the Lebesgue measure on V ) of the radial
Dunkl process is written as [17]
pk
t (x, y) =
1
cktγ+m/2
e−(|x|2+|y|2)/2tDW
k
(
x√
t
,
y√
t
)
ωk(y)2 (1)
for x, y ∈ C, where
γ =
∑
α∈R+
k(α), ωk(y) =
∏
α∈R+
〈α, y〉k(α).
The kernel DW
k is defined by
DW
k (x, y) =
∑
w∈W
Dk(x,wy)
where Dk is the non symmetric Dunkl kernel [18], and is up to the constant factor 1/|W | the
so-called generalized Bessel function since it reduces to the (normalized) Bessel function in the
rank-one case B1 [18]. It was identified in [1] with a multivariate hypergeometric series for both
root systems of types A and B. Since the C-type root system is nothing but the dual root
system of B, it remains to write down this kernel for the root system of type D which is the
remaining infinite family of irreducible root systems associated with Weyl groups. This was
behind our motivation to write the present paper and our main result is stated as
Theorem 1. For the D-type root system, the generalized Bessel function writes
1
|W |
∑
w∈W
Dk(x,wy) =
m∏
i=1
(xiyi
2
)
0F
(1/k1)
1
(
q +
1
2
,
x2
2
,
y2
2
)
+ 0F
(1/k1)
1
(
q − 1
2
,
x2
2
,
y2
2
)
,
where 0F
(1/k1)
1 is the multivariate hypergeometric series of Jack parameter 1/k1, x := (x2
1, . . . ,
x2
m), k = k1 > 0 is the value of the multiplicity function and q := 1 + (m− 1)k1.
Our strategy is easily explained as follows: according to Grabiner [9, p. 21], the Brownian
motion in the Weyl Chamber is a radial Dunkl process of multiplicity function k ≡ 1, that is
k(α) = 1 for all α ∈ R. Indeed, it is shown in [19] that the radial Dunkl process, say XW , is the
unique strong solution of
dYt = dBt +
∑
α∈R+
k(α)dt
〈α, Yt〉
, Y0 ∈ C,
where B is a m-dimensional Brownian motion and k(α) > 0 for all α ∈ R. Hence, by Grabiner’s
result on Brownian motions started at C and killed when they first reach ∂C together with the
Doob’s transform [16] applied with the harmonic function [9, p. 19]
h(y) :=
∏
α∈R+
〈α, y〉,
the semi group density the Brownian motion in the Weyl chamber is given by
p1
t (x, y) =
h(y)
h(x)
∑
w∈W
det(w)Nt,m(wy − x), (2)
Generalized Bessel function of Type D 3
where Nt,m is the m-dimensional heat kernel given by
Nt,m(x) =
1
(2πt)m/2
e−|x|
2/2t, |x|2 = 〈x, x〉, x ∈ V.
On the one hand and for the irreducible root systems A, B, D, the sum over W in (2) was ex-
pressed in [9] as determinants. On the other hand, the generalized Bessel function was expressed
for A and B-types root systems via multivariate hypergeometric series of two arguments and of
Jack parameter which equals the inverse of one of the multiplicity values (see the end of [1]).
When this parameter equals one, the multivariate series takes a determinantal form [10] which
agrees with Grabiner’s result. As a matter of fact, we will start from the expression obtained
in [9] for the Brownian motion in the Weyl chamber of type D then express it as a multivariate
series of a Jack parameter 1. Once we did, we will generalize the result to an arbitrary positive
Jack parameter using an analytical tool known as the shift principle. Before proceeding, we
shall recall some facts on multivariate hypergeometric series and investigate the cases of A and
B-types root systems.
2 Mutlivariate series and determinantal representations
We refer the reader to [1, 2, 12] and references therein for facts on Jack polynomials and multi-
variate hypergeometric series. Let τ be a partition of length m, that is a sequence of positive
integers τ1 > · · · > τm. Let α > 0, then the Jack polynomial C
(α)
τ of Jack parameter α is the
unique (up to a normalization) symmetric homogeneous eigenfunction of the operator
m∑
i=1
x2
i ∂
2
i +
2
α
m∑
i=1
x2
i
xi − xj
∂i
corresponding to the eigenvalue
ρτ =
m∑
i=1
τi[τi − (2/α)(i− 1)] + |τ |(m− 1), |τ | = τ1 + · · ·+ τm.
The normalization we adopt here is
(x1 + · · ·+ xm)n =
∑
τ
C(α)
τ (x), n ≥ 0,
where the sum is taken over all the partitions of weight |τ | = n and length m. For α = 2
it reduces to the zonal polynomial [15] while for α = 1 it fits the Schur polynomial [14]. Let
p, q ∈ N, then the multivariate hypergeometric series of two arguments is defined by
pF
(α)
q ((ai)1≤i≤p, (bj)1≤j≤q;x, y) =
∞∑
n=0
∑
|τ |=n
(a1)τ · · · (ap)τ
(b1)τ · · · (bq)τ
C
(α)
τ (x)C(α)
τ (y)
C
(α)
τ (1)|τ |!
,
where (1) = (1, . . . , 1) and for a partition τ
(a)(α)
τ =
m∏
i=1
(
a− 1
α
(i− 1)
)
τj
=
m∏
i=1
Γ(a− (i− 1)/α + τj)
Γ(a− (i− 1)/α)
is the generalized Pochhammer symbol and Γ is the Gamma function. We assume of course that
the above expression makes sense for all the coefficients (ai)1≤i≤p, (bj)1≤j≤q. An interesting
4 N. Demni
feature of the hypergeometric series of Jack parameter α = 1 is that they are expressed through
univariate hypergeometric functions pFq [13] as follows [10]
pF
(1)
q ((m + µi)1≤i≤p, (m + φj)1≤j≤q;x, y) = π
m(m−1)
2
(p−q−1+1/α)
m∏
i=1
(m− 1)!
×
p∏
i=1
(Γ(µi + 1))m
Γ(m + µi)
q∏
j=1
Γ(m + φj)
(Γ(φj + 1))m
det
[
pFq((µi + 1)1≤i≤p, (1 + φj)1≤j≤q;xlyr)
]m
l,r=1
V (x)V (y)
for all µi, φj > −1, where V stands for the Vandermonde polynomial and empty product are
equal 1. For instance, when p = q = 0, we have
0F
(1)
0 (x, y) = π−
m(m−1)
2
(−1+1/α)
det(exiyj )m
i,j=1
V (x)V (y)
,
and for p = 0, q = 1, we similarly get
0F
(1)
1 (m + φ;x, y) = π
m(m−1)
2α
m∏
i=1
(m− 1)!
Γ(m + φ)
(Γ(φ + 1))m
det
[
0F1(1 + φ;xiyj)
]m
i,j=1
V (x)V (y)
for φ > −1.
3 Brownian motions in Weyl chambers: types A and B revisited
Brownian motions in Weyl chambers were deeply studied in [9] and they are shown to be h-
transforms in Doob’s sense of m independent real BMs killed when they first hit ∂C. They are
then interpreted as m independent particles constrained to stay in the Weyl chamber C. The
proofs of those properties use probabilistic arguments. Here, we show how, for both types A
and B, these properties follow from the above determinantal representation of multivariate
hypergeometric functions of two arguments. Let us first recall that the A-type root system is
defined by
Am−1 = {±(ei − ej), 1 ≤ i < j ≤ m},
with positive and simple systems given by
R+ = {ei − ej , 1 ≤ i < j ≤ m}, S = {ei − ei+1, 1 ≤ i ≤ m− 1},
where (ei)1≤i≤m is the standard basis of Rm, W being the permutation group. In this case,
V = Rm, the span of R is the hyperplane of Rm consisting of vectors whose coordinates sum to
zero and C = {x ∈ Rm, x1 > · · · > xm}. Besides, there is only one orbit so that k(α) := k1 ≥ 0
and the function h (product of the positive roots) is given by the Vandermonde polynomial.
Next, it was shown in [1] that DW
k is given up to the constant 1/|W | by 0F
(1/k1)
0 (x, y) (we
use another normalization than the one used in [1] so that the factor
√
2 is removed). Thus,
when k1 = 1, (1) writes for R = Am−1
p1
t (x, y) =
V (y)
V (x)
det(Nt(yj − xi))m
i,j=1, (3)
where Nt is the heat kernel, namely
Nt(v) =
1√
2πt
e−v2/2t.
Generalized Bessel function of Type D 5
Thus, the V -transform property is easily seen. A similar result holds for R = Bm. This root
system has the following data
R = {±ei, ±ei ± ej , 1 ≤ i < j ≤ m}, R+ = {ei, 1 ≤ i ≤ m, ei ± ej , 1 ≤ i < j ≤ m},
S = {ei − ei+1, 1 ≤ i ≤ m, em}, C = {y ∈ Rm, y1 > y2 > · · · > ym > 0}.
W is generated by transpositions and sign changes (xi 7→ −xi) and there are two orbits so that
k = (k0, k1) thereby γ = mk0 + m(m − 1)k1. The generalized Bessel function1 is given by [1,
p. 214]
1
|W |
DW
k (x, y) = 0F
(1/k1)
1
(
k0 + (m− 1)k1 +
1
2
,
x2
2
,
y2
2
)
.
It follows that
pk0,1
t (x, y) = C(m)
h(y)
h(x)
e−(|x|2+|y|2)/2t
tm/2
m∏
i,j=1
(xiyj
t
)
det
[
0F1
(
k0 +
1
2
,
(xiyj)2
4t2
)]n
i,j=1
,
where h(y) = V (y2)
n∏
i=1
yi and C(m) is a constant depending on m. Taking k0 = 1 and using
the identity [13]
0F1
(
3
2
, z
)
=
C
2
√
z
sinh
(
2
√
z
)
,
for some constant C, one gets for the Brownian motion in the B-type Weyl chamber
p1,1
t (x, y) =
h(y)
h(x)
det [Nt(yj − xi)−Nt(yj + xi)]
m
i,j=1 . (4)
The h-transform property is then obvious.
4 Generalized Bessel function of type D
The root system of type D is defined by [11, p. 42]
R = {±ei ± ej , 1 ≤ i < j ≤ m}, R+ = {ei ± ej , 1 ≤ i < j ≤ m},
and there is one orbit so that k(α) = k1 thereby γ = m(m − 1)k1. The Weyl chamber is given
by:
C = {x ∈ Rm, x1 > x2 > · · · > |xm|} = C1 ∪ smC1,
where C1 is the Weyl chamber of type B and sm stands for the reflection with respect to the
vector em acting by sign change on the variable xm. Now, Grabiner’s result reads in this case
p1
t (x, y) =
V (y2)
V (x2)
det[Nt(yi − xj)−Nt(yi + xj)]mi,j=1 + det[Nt(yi − xj) + Nt(yi + xj)]mi,j=1
2
=
Cm
tγ+m/2
e−(|x|2+|y|2)/2t
det [sinh(xiyj/t)]mi,j=1 + det [cosh(xiyj/t)]mi,j=1
V (x2/4t2)V (y2)
V 2(y2),
1There is an erroneous sign in one of the arguments in [1]. Moreover, to recover this expression in the Bm case
from that given in [1], one should make substitutions a = k0 − 1/2, k1 = 1/α, q = 1 + (m− 1)k1.
6 N. Demni
where γ = m(m− 1). With the help of the determinantal representations and of [13]
0F1
(
3
2
, z
)
=
C√
z
sinh
(
2
√
z
)
, 0F1
(
1
2
, z
)
= cosh
(
2
√
z
)
,
one gets
p1
t (x, y) =
e−(|x|2+|y|2)/2t
cktγ+m/2
×
[
m∏
i=1
(xiyi
2t
)
0F
(1)
1
(
m +
1
2
,
x2
2t
,
y2
2t
)
+ 0F
(1)
1
(
m− 1
2
,
x2
2t
,
y2
2t
)]
V 2(y2).
With regard to (1) and setting q := 1+(m−1)k1, it is natural to prove the claim of Theorem 1.
Proof. It uses the so-called shift principle that we briefly outline [4]. Let E be a conjugacy
class of roots of R under the action W . Let kE be the value of the multiplicity function on this
class. Then, the generalized Bessel function associated with the multiplicity function k′ defined
by
k′(α) =
{
k(α) + 1 if α ∈ E,
k(α) otherwise,
is given by
1
|W |
DW
k′ (x, y) = C
∑
w∈W
ξE(w)
Dk(x,wy)
pE(x)pE(y)
, pE(x) :=
∏
α∈R+∩E
〈α, x〉,
where ξE(w) is defined by pE(wx) = ξE(w)pE(x) and C is some constant. Since we will deal with
both the B and D-types, it is convenient to add a superscript B or D to each corresponding item.
Therefore, WB, WD denote the Weyl groups associated with the root systems of types B, D
respectively and kB, kD denote the corresponding multiplicity functions. Recall that [11] WB is
the semi-direct product of Sm and (Z/2Z)m (sign changes) while WD is the semi-direct product
of Sm and (Z/2Z)m−1 (even sign changes). Let E := {ei, 1 ≤ i ≤ m}, then pE(x) is invariant
under permutations and even sign changes and skew-invariant under odd sign changes (note
that DW D
k is not WB-invariant). It follows that ξE(w) = 1 for w ∈ WD while ξE(w) = −1 for
w ∈ WB \WD. Now, recall that kD takes only one value while kB takes two values. Since the
Dunkl Laplacian ∆B
k of B reduces to ∆D
k when the value of the multiplicity function k0 on E is
zero [18], the (non-symmetric) Dunkl kernel DD
k associated with R = Dm is given by the Dunkl
kernel DB
k associated with R = Bm when kB
E = 0. In fact, this is true since Dk is the solution
of a spectral problem which is independent of W . As a result,
1
|WD|
DW D
k (x, y) =
1
|W |D
∑
w∈W D
DD
k (x, wy)
=
1
2|W |D
∑
w∈W B
(1 + ξE(w))DB,k0=0
k (x, wy)
=
|WB|
2|WD|
[
1
|WB|
D
W B(k0=0)
k +
pE(x)pE(y)
C|WB|
D
W B(k0=1)
k′
]
,
where we used the shift principle to derive the last line. Keeping in mind that
1
|WB|
DW B
k (x, y) = 0F
(1/k1)
1
(
k0 + (m− 1)k1 +
1
2
,
x2
2t
,
y2
2t
)
and using that |WB| = 2|WD| [11, p. 44] we are done with C = 2m. �
Generalized Bessel function of Type D 7
5 Concluding remarks
As the reader can see, DW D
k is not equal to DW B
k specialized with k0 = 0 and this shows that
the generalized Bessel function is intimately related to W . Moreover the symmetrical of DW D
k
with respect to sm gives DW B
k in the special case k0 = 0 which reflects the fact that the Weyl
chamber of type D is the union of the one of type B and its symmetrical with respect to sm.
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http://arxiv.org/abs/solv-int/9608004
http://arxiv.org/abs/0808.3033
http://arxiv.org/abs/math.PR/0609679
http://arxiv.org/abs/math.RT/9708207
http://arxiv.org/abs/math.CA/0210366
1 Root systems and related processes
2 Mutlivariate series and determinantal representations
3 Brownian motions in Weyl chambers: types A and B revisited
4 Generalized Bessel function of type D
5 Concluding remarks
References
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