## Symmetry, Integrability and Geometry: Methods and Applications Generalized Bessel function of Type D ⋆

### BibTeX

@MISC{Demni_symmetry,integrability,

author = {Nizar Demni},

title = {Symmetry, Integrability and Geometry: Methods and Applications Generalized Bessel function of Type D ⋆},

year = {}

}

### OpenURL

### Abstract

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 〈α, x〉 σα(x) = x − 2 α, x ∈ V. 〈α, α〉

### Citations

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Aspects of Multivariate Statistical Theory
- Muirhead
- 1982
(Show Context)
Citation Context ...ormalization 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) = ∞∑ where (1) = (1, ... |

453 |
Reflection groups and Coxeter groups, Cambridge
- Humphreys
- 1990
(Show Context)
Citation Context ...ambers; 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, ... |

347 |
Special functions and their applications
- Lebedev
- 1965
(Show Context)
Citation Context ...fficients (ai)1≤i≤p, (bj)1≤j≤q. An interesting (y) ,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) (p−q−1+1/α) 2 × p∏ i=1 (Γ(µi + 1)) m Γ(m + µi) q∏ j=1 Γ(m + φj) (Γ(φj + 1)) m m∏ (m − 1)! i=1 det [ pFq((µi + 1)1≤i≤p, (1 + φj... |

69 | The Calogero-Sutherland model and generalized classical polynomials
- BAKER, FORRESTER
- 1997
(Show Context)
Citation Context ... 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 r... |

66 | Brownian motion in a Weyl chamber, non-colliding particles, and random matrices
- Grabiner
- 1999
(Show Context)
Citation Context ...s 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 expressed 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 eq... |

55 |
Selberg integrals and hypergeometric functions associated with Jack polynomials
- Kaneko
- 1993
(Show Context)
Citation Context ...e 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 multivariate hypergeometric series. Let τ be a partition of length m, that is a sequence of positive integers τ1 > · · · > τm. Let α > 0, then... |

27 | Markov processes related with Dunkl operator
- RÖSLER, VOIT
- 1998
(Show Context)
Citation Context ...mental 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... |

25 | Certain hypergeometric series related to the root system - Beerends, Opdam - 1993 |

20 | Dunkl operators: theory and applications
- Rösler
(Show Context)
Citation Context ... ( x√t , y √ t ) ωk(y) 2 (1) for x, y ∈ C, where γ = ∑ k(α), α∈R+ ωk(y) = ∏ α∈R+ 〈α, y〉 k(α) . The kernel D W k D W k is defined by ∑ (x, y) = Dk(x, wy) w∈W 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 m... |

14 | Intertwining operators associated to the group S3
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(Show Context)
Citation Context ... ) x2 y2 , , 2t 2t ] V 2 (y 2 ). 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 f... |

12 |
Some new examples of Markov processes which enjoy the timeinversion property, Probab
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(Show Context)
Citation Context ... 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 C 2 c (C) as Lku(x)... |

8 |
M.: Some remarkable properties of the Dunkl martingales
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(Show Context)
Citation Context ... 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 C 2 c (C) as Lku(x)... |

5 | A chaotic representation property of the multidimensional Dunkl processes Ann. Probab
- Gallardo, Yor
(Show Context)
Citation Context ... 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 C 2 c (C) as Lku(x)... |

2 |
Skew-product representations of multidimensional Dunkl–Markov processes
- Chybiryakov
(Show Context)
Citation Context ... 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 C 2 c (C) as Lku(x) = 1 ∑ 〈α, ∇u(x)〉 ∆u... |

2 |
An invariance principle related to a process which generalizes N-dimensional Brownian motion
- Gallardo, Godefroy
(Show Context)
Citation Context |

2 |
D.St.P.: Total positivity, spherical series, and hypergeometric functions of matrix argument
- Gross, Richards
- 1989
(Show Context)
Citation Context ...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 ser... |

1 |
Symmetric functions and
- Mcdonald
- 1995
(Show Context)
Citation Context ...= ∑ 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) = ∞∑ where (1) = (1, . . . , 1) and for a partition τ (a) (α) τ = m∏ ( ... |

1 |
The Heckman–Opdam Markov processes, Probab. Theory Related Fields 138
- Schapira
- 2007
(Show Context)
Citation Context ... 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 X W , 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 ... |