Based on the theory of Dunkl operators, this paper presents a general concept of multivariable Hermite polynomials and Hermite functions which are associated with finite reflection groups on R N . The definition and properties of these generalized Hermite systems extend naturally those of their classical counterparts; partial derivatives and the usual exponential kernel are here replaced by Dunkl operators and the generalized exponential kernel K of the Dunkl transform. In case of the symmetric group SN , our setting includes the polynomial eigenfunctions of certain Calogero-Sutherland type operators. The second part of this paper is devoted to the heat equation associated with Dunkl's Laplacian. As in the classical case, the corresponding Cauchy problem is governed by a positive one-parameter semigroup; this is assured by a maximum principle for the generalized Laplacian. The explicit solution to the Cauchy problem involves again the kernel K; which is, on the way, proven to be nonn...

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Introduction The Schrodinger operator + fi(fi=2 \Gamma 1) ; 0 x j L (1.1) describes quantum particles on a line of length L interacting through a 1=r pair potential with periodic boundary conditions, or equivalently quantum particles on a circle of circumference length L (hence the superscript (C)) with the pair potential proportional to the inverse square of the chord length. It is one of a number of quantum many body systems in one dimension which are of the Calogero-Sutherland type, meaning that the ground state (i.e. eigenstate with the smallest eigenvalue E 0 ) is of the form e \Gammafi W=2 W consisting of one and two body terms only. Explicitly, for (1.1) we have W = W log je 2ix k =L \Gamma e j: (1.2) email: tbaker@maths.mu.oz.au email: matpjf@maths.mu.oz.au In studying the integrability properties of (1.1) it is useful [1] to generalize the Schrodinger operator to include the exchange operator M jk for coordinates x j and x k :

We investigate common algebraic structure for the rational and trigonometric Calogero-Sutherland models by using the exchange-operator formalism. We show that the set of the Jack polynomials whose arguments are Dunkl-type operators provides an orthogonal basis for the rational case. One dimensional quantum integrable models with long-range interaction have attracted much interest, because of not only their physical significance, but also their beautiful mathematical structure. One of such models is the Sutherland (trigonometric) model, which describes interacting particles on a circle[1]. The total momentum and Hamiltonian of the model are respectively given by N ∑ 1 ∂ Ps =, j=1 i ∂θj

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Summary. 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 heat semigroup, and related probabilistic aspects. The notes conclude with recent results on the asymptotics of the Dunkl kernel. 1 Introduction.............................................

In this paper we study the asymptotic behavior of the Jack rational functions P (z 1 ; : : : ; zn ; `) as the number of variables n and the signature grow to infinity. Our results generalize the results of A. Vershik and S. Kerov [VK2] obtained in the Schur function case (` = 1). For ` = 1=2; 2 our results describe approximation of the spherical functions of the infinite-dimensional symmetric spaces U(1)=O(1) and U(21)=Sp(1) by the spherical functions of the corresponding finite-dimensional symmetric spaces. Contents 1.1. Statement of the main result 1.2. Regular and infinitesimally regular sequences 1.3. Extremality of the limit functions 1.4. Related results 1.5. Acknowledgments 2. Jack polynomials and shifted Jack polynomials 2.1. Orthogonality 2.2. Interpolation 2.3. Branching rules 2.4. Binomial formula 2.5. Generating functions 2.6. Partitions and signatures 2.7. Extended symmetric functions 3. Asymptotic properties of Vershik-Kerov sequences of signatures 4. Sufficient conditions of regularity 5. Necessary conditions of regularity 5.1. The "only if " part of Theorem 1.1 5.2. A growth estimate for jf()j, f 2 7. Appendix. A direct proof of the formula (2.10) for generating functions The authors were supported by the Russian Basic Research Foundation grant 95-01-00814. The first author's stay at IAS in Princeton and MSRI in Berkeley was supported by NSF grants DMS--9304580 and DMS--9022140 respectively. Typeset by A M S-T E X 1 A. OKOUNKOV AND G. OLSHANSKI 1.1 Statement of the main result. Jack symmetric functions P (z 1 ; : : : ; z n ; `) 2 Q(`)[z \Sigma1 S(n) which are indexed by decreasing sequences of integers (called signatures) = ( 1 \Delta \Delta \Delta n ) 2 Z are eigenfunctions of the quantum Calogero-Sutherland Hamiltonian [C,Su] (1.1)...

In this paper, the distributions of the largest and smallest eigenvalues of complex Wishart matrices and the condition number of complex Gaussian random matrices are derived. These distributions are represented by complex hypergeometric functions of matrix arguments, which can be expressed in terms of complex zonal polynomials. Several results are derived on complex hypergeometric functions and complex zonal polynomials and are used to evaluate these distributions. Finally, applications of these distributions in numerical analysis and statistical hypothesis testing are mentioned.

ABSTRACT. We present an asymptotic expansion for quaternionic selfadjoint matrix integrals. The Feynman diagrams appearing in the expansion are ordinary ribbon graphs and their non-orientable counterparts. We show that the 2N × 2N Gaussian Orthogonal Ensemble (GOE) and N × N Gaussian Symplectic Ensemble (GSE) have exactly the same expansion term by term, except that the contributions from graphs on a non-orientable surface with odd Euler characteristic carry the opposite sign. As an application, we give a new topological proof of the known duality for correlations of characteristic polynomials, demonstrating that this duality is equivalent to Poincaré duality of graphs drawn on a compact surface. Another consequence of our graphical expansion formula is a simple and simultaneous (re)derivation of the Central Limit Theorem for GOE, GUE (Gaussian Unitary Ensemble) and GSE: The three cases have exactly the same graphical limiting formula except for an overall

In this paper we present a Maple library (MOPs) for computing Jack, Hermite, Laguerre, and Jacobi multivariate polynomials, as well as eigenvalue statistics for the Hermite, Laguerre, and Jacobi ensembles of Random Matrix theory. We also compute multivariate hypergeometric functions, and offer both symbolic and numerical evaluations for all these quantities. We prove that all algorithms are well-defined, analyze their complexity, and illustrate their performance in practice. Finally, we also present a few of the possible applications of this library.