A (p,q)-extension of Srivastava's triple hypergeometric function H_B and its properties

In this paper, we obtain a (p,q)-extension of Srivastava's triple hypergeometric function HB(⋅), by using the extended Beta function Bp,q(x,y) introduced by Choi et al. (2014). We give some of the main properties of this extended function, which include several integral representations involving Exton's hypergeometric function, the Mellin transform, a differential formula, recursion formulas and a bounded inequality. In addition, a new integral representation of the extended Srivastava triple hypergeometric function involving Laguerre polynomials is obtained.</p

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

Distributions on partitions, point processes, and the hypergeometric kernel

We study a 3-parametric family of stochastic point processes on the one-dimensional lattice originated from a remarkable family of representations of the infinite symmetric group. We prove that the correlation functions of the processes are given by determinantal formulas with a certain kernel. The kernel can be expressed through the Gauss hypergeometric function; we call it the hypergeometric kernel. In a scaling limit our processes approximate the processes describing the decomposition of representations mentioned above into irreducibles. As we showed before, see math.RT/9810015, the correlation functions of these limit processes also have determinantal form with so-called Whittaker kernel. We show that the scaling limit of the hypergeometric kernel is the Whittaker kernel. The integral operator corresponding to the Whittaker kernel is an integrable operator as defined by Its, Izergin, Korepin, and Slavnov. We argue that the hypergeometric kernel can be considered as a kernel defining a `discrete integrable operator'. We also show that the hypergeometric kernel degenerates for certain values of parameters to the Christoffel-Darboux kernel for Meixner orthogonal polynomials. This fact is parallel to the degeneration of the Whittaker kernel to the Christoffel-Darboux kernel for Laguerre polynomials.Comment: AMSTeX, 24 page