Q-special Functions Research Articles

Some limit relations in the A1 tableau of Dunkl-Cherednik operators

In this paper we study some limit relations involving some q-special functions related with the A 1 (root system) tableau of Dunkl-Cherednik operators. Concretely we consider the limits involving the nonsymmetric q-ultraspherical polynomials (q-Rogers polynomials), ultraspherical polynomials (Gegenbauer polynomials), q-Hermite and Hermite polynomials.

A Method for q-Calculus

We present a notation for q-calculus, which leads to a new method for computations and classifications of q-special functions. With this notation many formulas of q-calculus become very natural, and the q-analogues of many orthogonal polynomials and functions assume a very pleasant form reminding directly of their classical counterparts. The first main topic of the method is the tilde operator, which is an involution operating on the parameters in a q-hypergeometric series. The second topic is the q-addition, which consists of the Ward–AlSalam q-addition invented by Ward 1936 [102, p. 256] and Al-Salam 1959 [5, p. 240], and the Hahn q-addition. In contrast to the the Ward–AlSalam q-addition, the Hahn q-addition, compare [57, p. 362] is neither commutative nor associative, but on the other hand, it can be written as a finite product. We will use the generating function technique by Rainville [76] to prove recurrences for q-Laguerre polynomials, which are q-analogues of results in [76]. We will also find q-analogues of Carlitz’ [26] operator expression for Laguerre polynomials. The notation for Cigler’s [37] operational calculus will be used when needed. As an application, q-analogues of bilinear generating formulas for Laguerre polynomials of Chatterjea [33, p. 57], [32, p. 88] will be found.

Some Special Functions of Noncommuting Variables

In this paper we deal with q-commuting variables x and y satisfying the relation xy = qyx + (q − 1)y 2 with q complex, 0 < |q| < 1. We study various functional equations for q-exponentials and we deduce some identities for q-special functions involving q-commuting variab les.

Some Orthogonal Very-Well-Poised 8ϕ7-Functions that Generalize Askey–Wilson Polynomials

In a recent paper Ismail et al. (Algebraic Methods and q-Special Functions (J.F. van Diejen and L. Vinet, eds.) CRM Proceding and Lecture Notes, Vol. 22, American Mathematical Society, 1999, pp. 183–200) have established a continuous orthogonality relation and some other properties of a 2ϕ1-Bessel function on a q-quadratic grid. Dick Askey (private communication) suggested that the “Bessel-type orthogonality” found in Ismail et al. (1999) at the 2ϕ1-level has really a general character and can be extended up to the 8ϕ7-level. Very-well-poised 8ϕ7-functions are known as a nonterminating version of the classical Askey–Wilson polynomials (SIAM J. Math. Anal. 10 (1979), 1008–1016; Memoirs Amer. Math. Soc. Number319 (1985)). Askey's conjecture has been proved by the author in J. Phys. A: Math. Gen. 30 (1997), 5877–5885. In the present paper which is an extended version of Suslov (1997) we discuss in detail properties of the orthogonal 8ϕ7-functions. Another type of the orthogonality relation for a very-well-poised 8ϕ7-function was recently found by Askey et al. J. Comp. Appl. Math. 68 (1996), 25–55.

Bivariate Knop–Sahi and Macdonald polynomials related to q-ultraspherical functions

Knop and Sahi introduced a family of non-homogeneous and non-symmetric polynomials, G α ( x; q, t), indexed by compositions. An explicit formula for the bivariate Knop–Sahi polynomials reveals a connection between these polynomials and q-special functions. In particular, relations among the q-ultraspherical polynomials of Askey and Ismail, the two variable symmetric and non-symmetric Macdonald polynomials, and the bivariate Knop–Sahi polynomials are explicitly determined using the theory of basic hypergeometric series.

Intersection of modules related to Macdonald's polynomials

This work studies the intersection of certain k-tuples of Garsia-Haiman S n -modules M μ . We recall that in A. Garsia, M. Haiman, Electronic J. Combin. 3(2) Foata Festschrift (1996) R24, 60 for μ⊢n, M μ is defined as the linear span of all partial derivatives of a certain bihomogeneous polynomial Δ μ(X,Y) in the variables x 1,x 2,…,x n , y 1,y 2,…,y n . It has been conjectured that M μ has n! dimensions and that its bigraded Frobenius characteristic is given by a renormalized version of Macdonald's polynomials F. Bergeron, A. Garsia, Science fiction and Macdonald's polynomials, in: R. Floreanini, L. Vinet (Eds.), Algebraic Methods and q-Special Functions, CRM Proceedings & Lecture Notes, American Mathematical Society, Providence, RI, 48 pp. Computer data have suggested a precise presentation for certain irreducible representations of Frobenius characteristic S 2 k1 j appearing in M μ . This allows an explicit description of the intersection of M ν 's, as ν varies among immediate predecessors of a partition μ. We present here explicit results about the space ⋂ ν→μ M ν and its Frobenius characteristic, as well as a conjecture for the general form of this intersection. We give an explicit proof for hook shapes.

Unified approach to quasi-particle excitations in the edge states of paired Hall states

We consider the edge states of the quantum Hall paired states, which were introduced as the ground-state wave function for $\ensuremath{\nu}=\frac{5}{2}.$ We propose that the paired Hall states such as the Haldane-Rezayi state and the Pfaffian state have the supersymmetry, and we clarify the exclusion statistics of the quasi-particle excitations. By use of an idea of the ``restricted partition function'' for the quasi-particle excitations and its relationship with the q-special functions, we explicitly compute the distribution functions for the quasi-particle. We also calculate the specific heat and show that our results coincide with previously reported results based on the conformal field theory. We further show that there is a duality between the Pfaffian state and the Haldane-Rezayi state.

Functional equations with distortion and certain q-special functions

We show how some well-known q-special functions arise as solutions to analogue functional and difference equations using a doubly-indexed collection of algebras of formal power series type and the notion of distortion.

Quantum Group Approach to q-Special Functions

Quantum groups are used to define q-special functions. The Casimir operatorsof a variation of SUq(2) and Eq(2) are derived. The proposed q-associated Legendreand q-Bessel functions are the eigenfunctions of the Casimirs. The results differfrom ordinary q-special functions, but this is expected since the q-generalizationis not unique.

Special Functions of the Isomonodromy Type

We introduce a new notion, a special function of the isomonodromy type, and show that most of the functions known in applied mathematics and mathematical physics as special functions belong to this type. This definition provides a unified approach to the theories of ‘linear’ special functions, i.e., classical higher transcendental functions, and ‘nonlinear’ special functions, i.e., functions of the Painleve type. We also show that our definition has not only a conceptual (methodological) value; many well-known properties of the single-variable special functions can be re-derived not only from the isomonodromy point of view, but also the practical one too: (1) the isomonodromy approach is already known as an extremely useful one in the theory of Painleve functions, (2) many properties (some of them can be new ones) of the multi-variable special functions can be obtained on a regular basis, and (3) the definition gives rise to several interesting mathematical questions which are discussed in the paper. We also make some remarks concerning the analogous description (via q-deformations) of q-special functions.

Unitary representations of the quantum algebra suq(2) on a real two-dimensional sphere for q∈R+ or generic q∈S1

Some time ago, Rideau and Winternitz introduced a realization of the quantum algebra suq(2) on a real two-dimensional sphere, or a real plane, and constructed a basis for its representations in terms of q-special functions, which can be expressed in terms of q-Vilenkin functions, and are related to little q-Jacobi functions, q-spherical functions, and q-Legendre polynomials. In their study, the values of q were implicitly restricted to q∈R+. In the present paper, we extend their work to the case of generic values of q∈S1 (i.e., q values different from a root of unity). In addition, we unitarize the representations for both types of q values, q∈R+ and generic q∈S1, by determining some appropriate scalar products. From the latter, we deduce the orthonormality relations satisfied by the q-Vilenkin functions.

$q$ -deformed Hermite polynomials in $q$ -quantum mechanics

The q-special functions appear naturally in q-deformed quantum mechanics and both sides profit from this fact. Here we study the relation between the q-deformed harmonic oscillator and the q-Hermite polynomials. We discuss: recursion formula, generating function, Christoffel-Darboux identity, orthogonality relations and the moment functional.

{Quantum Algebras and q-Special Functions Related to Coherent States Maps of the Disc

The quantum algebras generated by the coherent states maps of the disc are investigated. It is shown that the analytic realization of these algebras leads to a generalized analysis which includes standard analysis as well as q-analysis. The applications of the analysis to star-product quantizations and q-special functions theory are given. Among others the meromorphic continuation of the generalized basic hypergeometric series is found and a reproducing measure is constructed, when the series is treated as a reproducing kernel.

Irreducible Representations of Deformed Oscillator Algebra and q-Special Functions

Different generators of a deformed oscillator algebra give rise to one-parameter families of q-exponential functions and q-Hermite polynomials. Connections of the Stieltjes and Hamburger classical moment problem with the corresponding resolution of unity are also pointed out.

Representations of the quantum algebra suq(2) on a real two-dimensional sphere

Representations of the quantum algebra suq(2) are constructed in terms of q-special functions, defined on real O(3) spheres and real planes. They include q-Vilenkin functions, related to little q-Jacobi functions, q-spherical functions, and q-Legendre polynomials.

Quantum Algebras and q-Special Functions

A quantum-algebraic framework for many q-special functions is provided. The two-dimensional Euclidean quantum algebra, sl q (2) and the q-oscillator algebra are considered. Realizations of these algebras in terms of operators acting on vector spaces of functions in one complex variable are given. Basic hypergeometric functions are shown to arise, in analogy with Lie theory, as matrix elements of certain operators. New generating functions for these q-special functions are obtained.

Quantum algebras and q-special functions: Roberto Floreanini. Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, Dipartimento di Fisica Teorica, Universita`di Trieste, Strada Costiera 11, 34014 Trieste, Italy; and Luc Vinet. Laboratoire de Physique Nucle´aire and Centre de Recherches Mathe´matiques, Universite´de Montre´al, Montre´al, Canada H3C 3J7

Quantum algebras and q-special functions: Roberto Floreanini. Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, Dipartimento di Fisica Teorica, Universita`di Trieste, Strada Costiera 11, 34014 Trieste, Italy; and Luc Vinet. Laboratoire de Physique Nucle´aire and Centre de Recherches Mathe´matiques, Universite´de Montre´al, Montre´al, Canada H3C 3J7