Q-hypergeometric Functions Research Articles

A unified approach to the summation and integration formulas for q-hypergeometric functions I

The most basic summation formula in the theory of q-hypergeometric functions is the well-known q-binomial formula. Not so well-known is the fact that there is a bilateral extension of it due to Ramanujan, and that there are two integral analogues of it. We show that these summation formulas as well as their integral counterparts have essentially the same origin, namely, a Pearson-type difference equation on a q-linear lattice. It is shown that the boundary conditions determine the structure of the solution of this equation which also enables us to evaluate the sums and integrals by a systematic process of iteration. We conclude by giving a very simple derivation of the q-Gauss formula and a second summation formula for a nonterminating 2ф 1 series.

Some Asymptotic Formulae for q -Hypergeometric Series

We obtain complete asymptotic expansions for certain q-hypergeometric series, including those pertaining to the Rogers–Ramanujan identities. Comparing asymptotic expansions associated with different q-series we discovered a new q-hypergeometric function identity.

Models ofQ-Algebra Representations: The Group of Plane Motions

This paper continues a study of one- and two-variable function space models of irreducible representations of q-analogs of Lie enveloping algebras, motivated by recurrence relations satisfied by q-hypergeometric functions. The algebra considered is the Lie algebra $m(2)$ of the group of plane motions. It is shown that various q-analogs of the exponential function can be used to mimic the exponential mapping from a Lie algebra to its Lie group, and the corresponding matrix elements of the group operators are computed on these representation spaces. This local approach applies to more general families of special functions, e.g., those with complex arguments and parameters, than does the quantum group approach. A simple one-variable model of the infinite-dimensional irreducible representations is used to compute the Clebsch–Gordan coefficients for $m(2)$ considered as a true quantum algebra. The authors derive a generalization of Koelink’s addition formula for Hahn–Exton q-Bessel functions. It is interpreted here as the expansion of the matrix elements of a group operator in a tensor product basis in terms of the matrix elements in a reduced basis.

Q-Hypergeometric functions, quantum algebras and free fields

We consider theq-analogsrφ of the generalized hypergeometric functionsrFs. Their free field realization and quantum algebraic interpretation are reviewed.

An exactly solvable class of discrete Schrodinger equations

A class of potentials admitting analytic solution of the discrete Schrodinger equation in terms of q-hypergeometric functions is considered. In particular, it includes the exponential and Hulthen potentials as well as the Maryland model.

Q -Hypergeometric Series and Macdonald Functions

We derive a duality formula for two-row Macdonald functions by studying their relation with basic hypergeometric functions. We introduce two parameter vertex operators to construct a family of symmetric functions generalizing Hall-Littlewood functions. Their relation with Macdonald functions is governed by a very well-poised q-hypergeometric functions of type 4Φ3, for which we obtain linear transformation formulas in terms of the Jacobi theta function and the q-Gamma function. The transformation formulas are then used to give the duality formula and a new formula for two-row Macdonald functions in terms of the vertex operators. The Jack polynomials are also treated accordingly.

Models of q-algebra representations: Matrix elements of the q-oscillator algebra

This article continues a study of function space models of irreducible representations of q analogs of Lie enveloping algebras, motivated by recurrence relations satisfied by q-hypergeometric functions. Here a q analog of the oscillator algebra (not a quantum algebra) is considered. It is shown that various q analogs of the exponential function can be used to mimic the exponential mapping from a Lie algebra to its Lie group and the corresponding matrix elements of the ‘‘group operators’’ on these representation spaces are computed. This ‘‘local’’ approach applies to more general families of special functions, e.g., with complex arguments and parameters, than does the quantum group approach. It is shown that the matrix elements themselves transform irreducibly under the action of the algebra. q analogs of a formula are found for the product of two hypergeometric functions 1F1 and the product of a 1F1 and a Bessel function. They are interpreted here as expansions of the matrix elements of a ‘‘group operator’’ (via the exponential mapping) in a tensor product basis (for the tensor product of two irreducible oscillator algebra representations) in terms of the matrix elements in a reduced basis. As a by-product of this analysis an interesting new orthonormal basis was found for a q analog of the Bargmann–Segal Hilbert space of entire functions.

Q-HYPERGEOMETRIC FUNCTIONS IN THE FORMALISM OF FREE FIELDS

We describe a representation of the q-hypergeometric functions of one variable in terms of correlators of vertex operators made out of free scalar fields on the Riemann sphere.

Models of q-algebra representations: Tensor products of special unitary and oscillator algebras

This paper begins a study of one- and two-variable function space models of irreducible representations of q analogs of Lie enveloping algebras, motivated by recurrence relations satisfied by q-hypergeometric functions. The algebras considered are the quantum algebra Uq(su2) and a q analog of the oscillator algebra (not a quantum algebra). In each case a simple one-variable model of the positive discrete series of finite- and infinite-dimensional irreducible representations is used to compute the Clebsch–Gordan coefficients. It is shown that various q analogs of the exponential function can be used to mimic the exponential mapping from a Lie algebra to its Lie group and the corresponding matrix elements of the ‘‘group operators’’ on these representation spaces are computed. It is shown that the matrix elements are polynomials satisfying orthogonality relations analogous to those holding for true irreducible group representations. It is also demonstrated that general q-hypergeometric functions can occur as basis functions in two-variable models, in contrast with the very restricted parameter values for the q-hypergeometric functions arising as matrix elements in the theory of quantum groups.

Q-conformal quantum mechanics and q-special functions

The q-difference analog of the most general conformal-invariant quantum hamiltonian is considered. It is seen to be one generator of the quantum algebra su q (1, 1). This is used to establish a connection between su q (1, 1) and q-hypergeometric functions, from which a new generating function for these q-series is obtained. The supersymmetric generalization is also briefly examined and a q-difference realization of osp q (1, 2) provided.

Ramanujan's q-hypergeometric functions in lattice statistics problems

Mathematical techniques leading to exact q-series type generating functions for several lattice and stacking models are presented. The structure of the generating function singularities is related to universality. These results have been obtained in collaboration with V. Privman and were extensively reviewed in Ref. 3.

Q-Series Identities and Reducibility of Basic Double Hypergeometric Functions

For real or complex q, |q| < 1, let(1.1)for arbitrary λ and μ, so that(1.2)and(1.3)Define, as usual, a generalized q-hypergeometric function by (cf. [26, Chapter 3]; see also [18])(1.4)where, for convergence, |q| < 1 and |z| < ∞ when r is a positive integer, or |z| < 1 when r = 0, provided that no zeros appear in the denominator.

Q-Hypergeometric Functions and Applications (H. Exton)

<i>q</i>-Hypergeometric Functions and Applications (H. Exton)

Macdonald’s polynomials and representations of quantum groups

In this paper we present a formula for Macdonald's polynomials for the root system A(n-1) which arises from the representation theory of quantum sl(n). This formula expresses Macdonald's polynomials via (weighted) traces of intertwining operators between certain modules over quantum sl(n). We also describe the commutative system of Macdonald's difference operators using the generators of the center of the quantum universal enveloping algebra, and use this description to prove a trace formula for generic eigenfunctions of these operators. These functions are generalized q-hypergeometric functions which are related to solutions of the quantum Knizhnik-Zamolodchikov equations.