Automorphisms of the q-deformed algebra suq(1, 1) and d-Orthogonal polynomials of q-Meixner type

Starting from an operator given as a product of q-exponential functions in irreducible representations of the positive discrete series of the q-deformed algebra su_q(1, 1), we express the associated matrix elements in terms of d-orthogonal polynomials. An algebraic setting allows to establish some properties : recurrence relation, generating function, lowering operator, explicit expression and d-orthogonality relations of the involved polynomials which are reduced to the orthogonal q-Meixner polynomials when d=1. If q ↑ 1, these polynomials tend to some d-orthogonal polynomials of Meixner type.     

Generalized q-Hermite Polynomials

We consider two operators A and A ⁺ in a Hilbert space of functions on the exponential lattice , where 0<q<1. The operators are formal adjoints of each other and depend on a real parameter . We show how these operators lead to an essentially unique symmetric ground state ψ₀ and that A and A ⁺ are ladder operators for the sequence . The sequence (ψ _n /ψ₀) is shown to be a family of orthogonal polynomials, which we identify as symmetrized q-Laguerre polynomials. We obtain in this way a new proof of the orthogonality for these polynomials. When γ=0 the polynomials are the discrete q-Hermite polynomials of type II, studied in several papers on q-quantum mechanics. Received: 6 December 1999 / Accepted: 21 May 2001     

Big q-Jacobi polynomials,q-Hahn polynomials,and a family of quantum 3-spheres

The big q-Jacobi polynomials and the q-Hahn polynomials are realized as spherical functions on a new quantum SU _q (2)-space which can be regarded as the total space of a family of quantum 3-spheres.     

Counting lattice animals: A parallel attack

A parallel algorithm for the enumeration of isolated connected clusters on a regular lattice is presented. The algorithm has been implemented on 17 RISC-based workstations to calculate the perimeter polynomials for the plane triangular lattice up to clustersizes=21. New data for perimeter polynomials D_s up toD ₂₁, total number of clustersg _s up tog ₂₂, and coefficientsb _r in the low-density series expansion of the mean cluster size up tob ₂₁ are given.     

Extended q -euler numbers and polynomials associated with fermionic p -adic q -integral on Z p

The purpose of this paper is to construct extended q-Euler numbers and polynomials related to fermionic p-adic q-integral on ℤ _p . By evaluating a multivariate p-adic q-integral on ℤ _p , we give new explicit formulas related to these numbers and polynomials.     

Quantum group invariants and link polynomials

A general method is developed for constructing quantum group invariants and determining their eigenvalues. Applied to the universalR-matrix this method leads to the construction of a closed formula for link polynomials. To illustrate the application of this formula, the quantum groupsU _q (E ₈),U _q (so(2m+1) andU _q (gl(m)) are considered as examples, and corresponding link polynomials are obtained.     

On integral and finite Fourier transforms of continuous <Emphasis Type="Italic">q</Emphasis>-Hermite polynomials

We give an overview of the remarkably simple transformation properties of the continuous q-Hermite polynomials H _n (x|q) of Rogers with respect to the classical Fourier integral transform. The behavior of the q-Hermite polynomials under the finite Fourier transform and an explicit form of the q-extended eigenfunctions of the finite Fourier transform, defined in terms of these polynomials, are also discussed. The text was submitted by the authors in English.     

q-Ultraspherical polynomials for q a root of unity

Properties of the q-ultraspherical polynomials for q being a primitive root of unity, are derived using a formalism of the so _q (3) algebra. The orthogonality condition for these polynomials provides a new class of trigonometric identities representing discrete finite-dimensional analogues of q-beta integrals of Ramanujan.     

Dirac(-Pauli), Fokker–Planck equations and exceptional Laguerre polynomials

An interesting discovery in the last two years in the field of mathematical physics has been the exceptional X_? Laguerre and Jacobi polynomials. Unlike the well-known classical orthogonal polynomials which start with constant terms, these new polynomials have lowest degree ? = 1, 2, … , and yet they form complete set with respect to some positive-definite measure. While the mathematical properties of these new X_? polynomials deserve further analysis, it is also of interest to see if they play any role in physical systems. In this paper we indicate some physical models in which these new polynomials appear as the main part of the eigenfunctions. The systems we consider include the Dirac equations coupled minimally and non-minimally with some external fields, and the Fokker–Planck equations. The systems presented here have enlarged the number of exactly solvable physical systems known so far.     

New Properties of the Zeros of Krall Polynomials

We identify a new class of algebraic relations satisfied by the zeros of orthogonal polynomials that are eigenfunctions of linear differential operators of order higher than two, known as Krall polynomials. Given an orthogonal polynomial family , we relate the zeros of the polynomial p_N with the zeros of p_m for each m ≤ N (the case m = N corresponding to the relations that involve the zeros of p_N only). These identities are obtained by finding exact expressions for the similarity transformation that relates the spectral and the (interpolatory) pseudospectral matrix representations of linear differential operators, while using the zeros of the polynomial p_N as the interpolation nodes. The proposed framework generalizes known properties of classical orthogonal polynomials to the case of nonclassical polynomial families of Krall type. We illustrate the general result by proving new identities satisfied by the Krall-Legendre, the Krall-Laguerre and the Krall-Jacobi orthogonal polynomials.     

Duality, Biorthogonal Polynomials¶and Multi-Matrix Models

The statistical distribution of eigenvalues of pairs of coupled random matrices can be expressed in terms of integral kernels having a generalized Christoffel–Darboux form constructed from sequences of biorthogonal polynomials. For measures involving exponentials of a pair of polynomials V ₁, V ₂ in two different variables, these kernels may be expressed in terms of finite dimensional “windows” spanned by finite subsequences having length equal to the degree of one or the other of the polynomials V ₁, V ₂. The vectors formed by such subsequences satisfy “dual pairs” of first order systems of linear differential equations with polynomial coefficients, having rank equal to one of the degrees of V ₁ or V ₂ and degree equal to the other. They also satisfy recursion relations connecting the consecutive windows, and deformation equations, determining how they change under variations in the coefficients of the polynomials V ₁ and V ₂. Viewed as overdetermined systems of linear difference-differential-deformation equations, these are shown to be compatible, and hence to admit simultaneous fundamental systems of solutions. The main result is the demonstration of a spectral duality property; namely, that the spectral curves defined by the characteristic equations of the pair of matrices defining the dual differential systems are equal upon interchange of eigenvalue and polynomial parameters. Received: 14 September 2001 / Accepted: 18 February 2002     

Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on ℤ p

The objective of the paper is to indicate a symmetry of the multivariate p-adic invariant integral on ℤ _p , which leads to a relation between the power sum polynomials and higher-order Euler polynomials. The present research has been conducted under the Research Grant of Kwangwoon University in 2008.     

Asymptotic Statistics of Zeroes for the Lamé Ensemble

The Lamé polynomials naturally arise when separating variables in Laplace's equation in elliptic coordinates. The products of these polynomials form a class of spherical harmonics, which are joint eigenfunctions of a quantum completely integrable (QCI) system of commuting, second-order differential operators P ₀=Δ, P ₁,…,P ^{N −1} acting on C ^∞(? ^N ). These operators naturally depend on parameters and thus constitute an ensemble. In this paper, we compute the limiting level-spacings distributions for the zeroes of the Lamé polynomials in various thermodynamic, asymptotic regimes. We give results both in the mean and pointwise, for an asymptotically full set of values of the parameters. Received: 17 January 2001 / Accepted: 14 May 2001     

Dual properties of orthogonal polynomials of discrete variables associated with the quantum algebra <Emphasis Type="Italic">U</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>(<Emphasis Type="Italic">su</Emphasis>(2))

We show that for every set of discrete polynomials y _n (x(s)) on the lattice x(s), defined on a finite interval (a, b), it is possible to construct two sets of dual polynomials z _k (ξ(t)) of degrees k = s-a and k = b-s-1. Here we do this for the classical and alternative Hahn and Racah polynomials as well as for their q-analogs. Also we establish the connection between classical and alternative families. This allows us to obtain new expressions for the Clerbsch-Gordan and Racah coefficients of the quantum algebra U _q (su(2)) in terms of various Hahn and Racah q-polynomials. Dedicated to the memory of our teacher and friend Arnold F. Nikiforov (18.11.1930–27.12.2005).     

q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients

A purpose of this paper is to present a systemic study of some families of multiple q-Bernoulli numbers and polynomials by using the multivariate q-Volkenborn integral (= p-adic q-integral) on ℤ _p . Moreover, the study of these higher-order q-Bernoulli numbers and polynomials implies some interesting q-analogs of Stirling number identities. This paper is supported by Jangjeon Research Institute for Mathematical Science (JRIMS-10R-2001).     

Lifting <Emphasis Type="Italic">q</Emphasis>-difference operators for Askey-Wilson polynomials and their weight function

We determine an explicit form of a q-difference operator that transforms the continuous q-Hermite polynomials H _n (x|q) of Rogers into the Askey-Wilson polynomials p _n (x; a, b, c, d|q) on the top level in the Askey q-scheme. This operator represents a special convolution-type product of four one-parameter q-difference operators of the form ɛ _q (c _q D _q ) (where c _q are some constants), defined as Exton’s q-exponential function ɛ _q (z) in terms of the Askey-Wilson divided q-difference operator D _q . We also determine another q-difference operator that lifts the orthogonality weight function for the continuous q-Hermite polynomialsH _n (x|q) up to the weight function, associated with the Askey-Wilson polynomials p _n (x; a, b, c, d|q).     

On the multiple <Emphasis Type="Italic">q</Emphasis>-Genocchi and Euler numbers

The purpose of this paper is to present a systematic study of some families of multiple q-Genocchi and Euler numbers by using the multivariate q-Volkenborn integral (= p-adic q-integral) on ℤ _p . The investigation of these q-Genocchi numbers and polynomials of higher order leads to interesting identities related to these objects. The results of the present paper cover earlier results concerning ordinary q-Genocchi numbers and polynomials. This paper is supported by Jangjeon Research Institute for Mathematical Science (JRIMS-11R-2007).     

Factorized Weight Functions vs. Factorized Scattering

Starting from an extensive class of factorized weight functions W(p) on the N-dimensional torus ?, we construct an orthonormal base of symmetric N-variable polynomials for L ² _s (?,W(p)dp) via lexicographic ordering of the monomial symmetric functions (free boson states) and the Gram-Schmidt procedure. We show that the dominant asymptotics of these polynomials is factorized. As a corollary, we obtain a large class of quantum integrable soliton systems on the symmetric subspace of l ²(ℤ ^N ). The class of weight functions contains in particular the weight function yielding Macdonald polynomials. For that special case, the quantum soliton system can be viewed as the dual relativistic Calogero–Sutherland system. Received: 4 September 2001 / Accepted: 4 January 2002     

Generalized Orthogonal Polynomials, Discrete KP and Riemann–Hilbert Problems

Classically, a single weight on an interval of the real line leads to moments, orthogonal polynomials and tridiagonal matrices. Appropriately deforming this weight with times t= (t ₁, t ₂, …), leads to the standard Toda lattice and τ-functions, expressed as hermitian matrix integrals. This paper is concerned with a sequence of t-perturbed weights, rather than one single weight. This sequence leads to moments, polynomials and a (fuller) matrix evolving according to the discrete KP-hierarchy. The associated τ-functions have integral, as well as vertex operator representations. Among the examples considered, we mention: nested Calogero–Moser systems, concatenated solitons and m-periodic sequences of weights. The latter lead to 2m+ 1-band matrices and generalized orthogonal polynomials, also arising in the context of a Riemann–Hilbert problem. We show the Riemann–Hilbert factorization is tantamount to the factorization of the moment matrix into the product of a lower–times upper–triangular matrix. Received: 8 September 1998 / Accepted: 27 April 1999     

Quantum 401-1401-1401-1algebras and Macdonald polynomials

We derive a quantum deformation of theW _N algebra and its quantum Miura transformation, whose singular vectors realize the Macdonald polynomials.