Terms Of Basic Hypergeometric Functions Research Articles

Nevanlinna extremal measures for polynomials related to q−1-Fibonacci polynomials

The aim of this paper is the study of q−1-Fibonacci polynomials with 0<q<1. First, the q−1-Fibonacci polynomials are related to a q-exponential function which allows an asymptotic analysis to be worked out. Second, related basic orthogonal polynomials are investigated with the emphasis on their orthogonality properties. In particular, a compact formula for the reproducing kernel is obtained that allows to describe all the N-extremal measures of orthogonality in terms of basic hypergeometric functions and their zeros. Two special cases involving q-sine and q-cosine are discussed in more detail.

Geometric properties of basic hypergeometric functions

In this article, we consider basic hypergeometric functions introduced by Heine. We study the mapping properties of certain ratios of basic hypergeometric functions having shifted parameters and show that they map the domains of analyticity onto domains convex in the direction of the imaginary axis. In order to investigate these mapping properties, some useful identities are obtained in terms of basic hypergeometric functions. In addition, we find conditions under which the basic hypergeometric functions are in a q-close-to-convex family.

Representations for the first associated q-classical orthogonal polynomials

Let {p k(x; q)} be any system of the q-classical orthogonal polynomials, and let ϱ be the corresponding weight function, satisfying the q-difference equation D q ( σϱ)= τϱ, where σ and τ are polynomials of degree at most 2 and exactly 1, respectively. Further, let { p k (1)( x; q)} be associated polynomials of the polynomials {p k(x; q)} . Explicit forms of the coefficients b n, k and c n, k in the expansions p n−1 (1)(x;q)= ∑ k=0 n−1 b n,kϑ k(x), p n−1 (1)(x;q)= ∑ k=0 n−1 c n,kp k(x; q) are given in terms of basic hypergeometric functions. Here ϑ k ( x) equals x k if σ +(0)=0, or ( x; q) k if σ +(1)=0, where σ +( x)≔ σ( x)+( q−1) xτ( x). The most important representatives of those two classes are the families of little q-Jacobi and big q-Jacobi polynomials, respectively. Writing the second-order nonhomogeneous q-difference equation satisfied by p n−1 (1)( x; q) in a special form, recurrence relations (in k) for b n, k and c n, k are obtained in terms of σ and τ.

Basic Hypergeometric Functions and the Borel–Weil Construction for $U_q (3)$

The Borel–Weil construction of the irreducible unitary representations of the quantum group $U_q (3)$ is investigated. The representation functions are calculated explicitly and found to be expressible in terms of basic hypergeometric functions; the form of these basis functions verifies previous general results. It is shown that several identities satisfied by basic hypergeometric functions, including a special case of Watson’s formula, are implied by properties of the quantum group.

QUANTUM DEFORMATIONS OF QUANTUM MECHANICS

Based on a deformation of the quantum mechanical phase space we study q-deformations of quantum mechanics for qk=1 and 0&lt;q&lt;1. After defining a q-analog of the scalar product on the function space we discuss and compare the time evolution of operators in both cases. A formulation of quantum mechanics for qk=1 is given and the dynamics for the free Hamiltonian is studied. For 0&lt;q&lt;1 we develop a deformation of quantum mechanics and the cases of the free Hamiltonian and the one with a x2-potential are solved in terms of basic hypergeometric functions.