PL EN

Preferencje
Język
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo

Colloquium Mathematicum

2011 | 123 | 2 | 149-179
Tytuł artykułu

Multidimensional Heisenberg convolutions and product formulas for multivariate Laguerre polynomials

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let p,q be positive integers. The groups $U_{p}(ℂ)$ and $U_{p}(ℂ) × U_q(ℂ)$ act on the Heisenberg group $H_{p,q}: = M_{p,q}(ℂ) × ℝ$ canonically as groups of automorphisms, where $M_{p,q}(ℂ)$ is the vector space of all complex p × q matrices. The associated orbit spaces may be identified with $Π_q × ℝ$ and $Ξ_q × ℝ$ respectively, $Π_q$ being the cone of positive semidefinite matrices and $Ξ_q$ the Weyl chamber ${x ∈ ℝ^q: x₁ ≥ ⋯ ≥ x_q ≥ 0}$. In this paper we compute the associated convolutions on $Π_q × ℝ$ and $Ξ_q × ℝ$ explicitly, depending on p. Moreover, we extend these convolutions by analytic continuation to series of convolution structures for arbitrary parameters p ≥ 2q-1. This leads for q ≥ 2 to continuous series of noncommutative hypergroups on $Π_q × ℝ$ and commutative hypergroups on $Ξ_q × ℝ$. In the latter case, we describe the dual space in terms of multivariate Laguerre and Bessel functions on $Π_q$ and $Ξ_q$. In particular, we give a nonpositive product formula for these Laguerre functions on $Ξ_q$. The paper extends the known case q = 1 due to Koornwinder, Trimèche, and others, as well as the group case with integers p due to Faraut, Benson, Jenkins, Ratcliff, and others. Moreover, our results are closely related to product formulas for multivariate Bessel and other hypergeometric functions of Rösler.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
149-179
Opis fizyczny
Daty
wydano
2011
Twórcy
autor
• Fakultät Mathematik, Technische Universität Dortmund, Vogelpothsweg 87, D-44221 Dortmund, Germany
Bibliografia
Typ dokumentu
Bibliografia
Identyfikatory