scholarly journals Heckman–Opdam's Jacobi polynomials for the BCn root system and generalized spherical functions

2004 ◽  
Vol 186 (1) ◽  
pp. 153-180 ◽  
Author(s):  
A Oblomkov
Keyword(s):  
Root System ◽  
Jacobi Polynomials ◽  
Spherical Functions

scholarly journals Spherical functions on orthogonal groups

1996 ◽  
Vol 141 ◽  
pp. 157-182 ◽  
Author(s):  
Yasuhiro Kajima
Keyword(s):  
Algebraic Group ◽  
Root System ◽  
Explicit Formula ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Spherical Functions ◽  
Orthogonal Groups ◽  
Reductive Algebraic Group

Let G be a p-adic connected reductive algebraic group and K a maximal compact subgroup of G. In [4], Casselman obtained the explicit formula of zonal spherical functions on G with respect to K on the assumption that K is special. It is known (Bruhat and Tits [3]) that the affine root system of algebraic group which has good but not special maximal compact subgroup is A1 C2, or Bn (n > 3), and all Bn-types can be realized by orthogonal groups. Here the assumption “good” is necessary for the Satake’s theory of spherical functions.

scholarly journals Sonine Formulas and Intertwining Operators in Dunkl Theory

2020 ◽  
Author(s):  
Margit Rösler ◽  
Michael Voit
Keyword(s):  
Root System ◽  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Jacobi Polynomials ◽  
Necessary Conditions ◽  
Integral Representations ◽  
Intertwining Operators ◽  
Dunkl Operators ◽  
Intertwining Operator ◽  
Type Integral

Abstract Let $V_k$ denote Dunkl’s intertwining operator associated with some root system $R$ and multiplicity $k$. For two multiplicities $k, k^{\prime }$ on $R$, we study the intertwiner $V_{k^{\prime },k} = V_{k^{\prime }}\circ V_k^{-1}$ between Dunkl operators with multiplicities $k$ and $k^{\prime }.$ It has been a long-standing conjecture that $V_{k^{\prime },k}$ is positive if $k^{\prime } \geq k \geq 0.$ We disprove this conjecture by constructing counterexamples for root system $B_n$. This matter is closely related to the existence of Sonine-type integral representations between Dunkl kernels and Bessel functions with different multiplicities. In our examples, such Sonine formulas do not exist. As a consequence, we obtain necessary conditions on Sonine formulas for Heckman–Opdam hypergeometric functions of type $BC_n$ and conditions for positive branching coefficients between multivariable Jacobi polynomials.

The Spherical Functions Related to the Root System B2

2002 ◽  
Vol 45 (3) ◽  
pp. 436-447
Author(s):  
P. Sawyer
Keyword(s):  
Root System ◽  
Differential Operators ◽  
Integral Formula ◽  
Spherical Functions

AbstractIn this paper, we give an integral formula for the eigenfunctions of the ring of differential operators related to the root system B2.

Limit Transitions for BC Type Multivariable Orthogonal Polynomials

1997 ◽  
Vol 49 (2) ◽  
pp. 374-405 ◽  
Author(s):  
Jasper V. Stokman ◽  
Tom H. Koornwinder
Keyword(s):  
Orthogonal Polynomials ◽  
Root System ◽  
Jacobi Polynomials ◽  
Parameter Family ◽  
Difference Operators ◽  
Wilson Polynomials ◽  
The One

AbstractLimit transitions will be derived between the five parameter family of Askey-Wilson polynomials, the four parameter family of big q-Jacobi polynomials and the three parameter family of little q-Jacobi polynomials in n variables associated with root system BC. These limit transitions generalize the known hierarchy structure between these families in the one variable case. Furthermore it will be proved that these three families are q-analogues of the three parameter family of BC type Jacobi polynomials in n variables. The limit transitions will be derived by taking limits of q-difference operators which have these polynomials as eigenfunctions.

scholarly journals Limit transition between hypergeometric functions of type BC and type A

2013 ◽  
Vol 149 (8) ◽  
pp. 1381-1400 ◽  
Author(s):  
Margit Rösler ◽  
Tom Koornwinder ◽  
Michael Voit
Keyword(s):  
Hypergeometric Functions ◽  
Grassmann Manifold ◽  
Jacobi Polynomials ◽  
Type A ◽  
Explicit Representation ◽  
Spherical Functions ◽  
Limit Transition ◽  
Infinite Dimensional ◽  
Positive Roots ◽  
General Limit

AbstractLet ${F}_{BC} (\lambda , k; t)$ be the Heckman–Opdam hypergeometric function of type BC with multiplicities $k= ({k}_{1} , {k}_{2} , {k}_{3} )$ and weighted half-sum $\rho (k)$ of positive roots. We prove that ${F}_{BC} (\lambda + \rho (k), k; t)$ converges as ${k}_{1} + {k}_{2} \rightarrow \infty $ and ${k}_{1} / {k}_{2} \rightarrow \infty $ to a function of type A for $t\in { \mathbb{R} }^{n} $ and $\lambda \in { \mathbb{C} }^{n} $. This limit is obtained from a corresponding result for Jacobi polynomials of type BC, which is proven for a slightly more general limit behavior of the multiplicities, using an explicit representation of Jacobi polynomials in terms of Jack polynomials. Our limits include limit transitions for the spherical functions of non-compact Grassmann manifolds over one of the fields $ \mathbb{F} = \mathbb{R} , \mathbb{C} , \mathbb{H} $ when the rank is fixed and the dimension tends to infinity. The limit functions turn out to be exactly the spherical functions of the corresponding infinite-dimensional Grassmann manifold in the sense of Olshanski.

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