The Dunkl operators associated to a dihedral group are a pair of differential-difference operators that generate a commutative algebra acting on differentiable functions in $\mathbb{R}^2$. The intertwining operator intertwines between this algebra and the algebra of differential operators. The main result of this paper is an integral representation of the intertwining operator on a class of functions. As an application, closed formulas for the Poisson kernels of $h$-harmonics and sieved Gegenbauer polynomials are deduced when one of the variables is at vertices of a regular polygon, and similar formulas are also derived for several other related families of orthogonal polynomials.
Dunkl operators associated with finite reflection groups generate a commutative algebra of differential-difference operators. There exists a unique linear operator called intertwining operator which intertwines between this algebra and the algebra of standard differential operators. There also exists a generalization of the Fourier transform in this context called Dunkl transform. In this paper, we determine an integral expression for the Dunkl kernel, which is the integral kernel of the Dunkl transform, for all dihedral groups. We also determine an integral expression for the intertwining ope…
There is a commutative algebra of differential-difference operators, with two parameters, associated to any dihedral group with an even number of reflections. The intertwining operator relates this algebra to the algebra of partial derivatives. This paper presents an explicit form of the action of the intertwining operator on polynomials by use of harmonic and Jacobi polynomials. The last section of the paper deals with parameter values for which the formulae have singularities.
In this paper, we establish an integral expression for the Dunkl kernel in the context of Dihedral group of an arbitrary order by using the results in \cite{M-Y-Vk} where a construction of the Dunkl intertwining operator for a large set of regular parameter functions is provided. We introduce a differential equations systems that leads to the explicit expression of the Dunkl Kernel whenever an appropriate solution of it is obtained. In particular, an explicit expression of the Dunkl kernel $E_k(x,y)$ is given when one of its argument $x$ or $y$ is invariant under the action of a known reflecti…
An integral representation of the intertwining operator for the Dunkl operators associated with symmetric groups is derived for the class of functions of a single component. The expression provides a closed form formula for the reproducing kernels of $h$-harmonics associated with symmetric groups when one of the components is a coordinate vector. The latter allows us to establish a sharp result for the Cesàro summability of $h$-harmonic series on the unit sphere.
The space of polynomials in two real variables with values in a 2-dimensional irreducible module of a dihedral group is studied as a standard module for Dunkl operators. The one-parameter case is considered (omitting the two-parameter case for even dihedral groups). The matrix weight function for the Gaussian form is found explicitly by solving a boundary value problem, and then computing the normalizing constant. An orthogonal basis for the homogeneous harmonic polynomials is constructed. The coefficients of these polynomials are found to be balanced terminating $_4F_3$-series.
Motivated by Dunkl operators theory, we consider a generating series involving a modified Bessel function and a Gegenbauer polynomial, that generalizes a known series already considered by L. Gegenbauer. We actually use inversion formulas for Fourier and Radon transforms to derive a closed formula for this series when the parameter of the Gegenbauer polynomial is a strictly positive integer. As a by-product, we get a relatively simple integral representation for the generalized Bessel function associated with even dihedral groups when both multiplicities sum to an integer. In particular, we re…
In this note, we express explicitly the Dunkl kernel and generalized Bessel functions of type $A_{n-1}$ by the Humbert's function $Φ_{2}^{(n)}$, with one variable specified. The obtained formulas lead to a new proof of Xu's integral expression for the intertwining operator associated to symmetric groups, which was recently reported in [21].
In this work, we consider the Dunkl complex reflection operators related to the group $G(m,1,N)$ in the complex plane \begin{align*} T_i=\frac{\partial}{\partial z_i}+k_0\sum_{j\neq i}\sum_{r=0}^{m-1}\frac{1-s_i^{-r}(i,j)s_i^r} {z_i-\varepsilon^r z_j}+\sum_{j=1}^{m-1}k_j\sum_{r=0}^{m-1}\frac{\varepsilon^{-rj}s_i^r}{z_i}, \,\,1\leq i\leq N. \end{align*} We first review the theory of Dunkl operators for complex reflection groups we recall some results related to the hyper--Bessel functions, which are solutions of a higher order differential equation. Secondly, we construct a new explicit intertw…
There is a commutative algebra of differential-difference operators, acting on polynomials on R_2, associated with the reflection group B2. This paper presents an integral transform which intertwines this algebra, allowing one free parameter, with the algebra of partial derivatives. The method of proof depends on properties of a certain class of balanced terminating hypergeometric series of 4F3-type. These properties are in the form of recurrence and contiguity relations and are proved herein.
In this paper, we pursue the investigations started in \cite{Mas-You} where the authors provide a construction of the Dunkl intertwining operator for a large subset of the set of regular multiplicity values. More precisely, we make concrete the action of this operator on homogeneous polynomials when the root system is of dihedral type and under a mild assumption on the multiplicity function. In particular, we obtain a formula for the corresponding Dunkl kernel and another representation of the generalized Bessel function already derived in \cite{Demni0}. When the multiplicity function is every…
In the first part of this paper, we express the generalized Bessel function associated with dihedral systems and a constant multiplicity function as a infinite series of confluent Horn functions. The key ingredient leading to this expression is an extension of an identity involving Gegenbauer polynomials proved in a previous paper by the authors, together with the use of the Poisson kernel for these polynomials. In particular, we derive an integral representation of this generalized Bessel function over the standard simplex. The second part of this paper is concerned with even dihedral systems…
In this paper, we introduce a new differential-difference operator $T_ξ$ $(ξ\in \mathbb{R}^N)$ by using projections associated to orthogonal subsystems in root systems. Similarly to Dunkl theory, we show that these operators commute and we construct an intertwining operator between $T_ξ$ and the directional derivative $\partial_ξ$. In the case of one variable, we prove that the Kummer functions are eigenfunctions of this operator.
Let $V_k$ denote Dunkl's intertwining operator associated with some root system $R$ and multiplicity function $k$. For two multiplicities $k, k^\prime$ on $R$, we study the operator $V_{k^\prime,k} = V_{k^\prime}\circ V_k^{-1}$, which intertwines the Dunkl operators for multiplicity $k$ with those for multiplicity $k^\prime.$ While it is well-known that the operator $V_k$ is positive for nonnegative $k$, it has been a long-standing conjecture that its generalizations $V_{k^\prime,k}$ are also positive if $k^\prime \geq k \geq 0,$ which is known to be true in rank one. In this paper, we disprov…
We consider the Dunkl intertwining operator $V_α$ and its dual ${}^tV_α$, we define and study the Dunkl Sonine operator and its dual on $\mathbb{R}$. Next, we introduce complex powers of the Dunkl Laplacian $Δ_α$ and establish inversion formulas for the Dunkl Sonine operator $S_{α,β}$ and its dual ${}^tS_{α,β}$. Also, we give a Plancherel formula for the operator ${}^tS_{α,β}$.
In this paper we consider the Bessel-Struve operator $l_α$ and the Bessel-Struve intertwining operator $χ_α$ and its dual, we define and study the Bessel-Struve Sonine transform $S_{α,β}$ on $\mathcal{E}(\mathbb{R})$. We prove that $S_{α,β}$ is a transmutation operator from $l_α$ into $l_β$ on $\mathcal{E}(\mathbb{R})$ and we deduce similar result for its dual $S_{α,β}^*$ on $\mathcal{E}'(\mathbb{R})$. Furthermore, invoking Weyl integral transform and the Dual Sonine transform $^tS_{α,β}$ on $\mathcal{D}(\mathbb{R})$, we get a relation between the Bessel-Struve transforms $\mathcal{F}^α_{BS} $…
These lecture notes are intended as an introduction to the theory of rational Dunkl operators and the associated special functions, with an emphasis on positivity and asymptotics. We start with an outline of the general concepts: Dunkl operators, the intertwining operator, the Dunkl kernel and the Dunkl transform. We point out the connection with integrable particle systems of Calogero-Moser-Sutherland type, and discuss some systems of orthogonal polynomials associated with them. A major part is devoted to positivity results for the intertwining operator and the Dunkl kernel, the Dunkl-type he…
We compute generalized Bernstein-Reznikov integrals associated with standard complex symplectic forms by studying Knapp-Stein intertwining operators between spherical degenerate principal series of complex symplectic groups.
In this paper, we are interested in a Neumann-type series for modified Bessel functions of the first kind which arises in the study of Dunkl operators associated with dihedral groups and as an instance of the Laguerre semigroup constructed by Ben Said-Kobayashi-Orsted. We first revisit the particular case corresponding to the group of square-preserving symmetries for which we give two new and different proofs other than the existing ones. The first proof uses the expansion of powers in a Neumann series of Bessel functions while the second one is based on a quadratic transformation for the Gaus…
Trigonometric formulas are derived for certain families of associated Legendre functions of fractional degree and order, for use in approximation theory. These functions are algebraic, and when viewed as Gauss hypergeometric functions, belong to types classified by Schwarz, with dihedral, tetrahedral, or octahedral monodromy. The dihedral Legendre functions are expressed in terms of Jacobi polynomials. For the last two monodromy types, an underlying `octahedral' polynomial, indexed by the degree and order and having a non-classical kind of orthogonality, is identified, and recurrences for it a…
In this paper, a family of radial deformations of the realization of the Lie superalgebra osp(1|2) in the theory of Dunkl operators is obtained. This leads to a Dirac operator depending on 3 parameters. Several function theoretical aspects of this operator are studied, such as the associated measure, the related Laguerre polynomials and the related Fourier transform. For special values of the parameters, it is possible to construct the kernel of the Fourier transform explicitly, as well as the related intertwining operator.
