Theq-harmonic oscillator and the Al-Salam and Carlitz polynomials

Abstract

Abstract. One more model of a q-harmonic oscillator based on the qorthogonal polynomials of Al-Salam and Carlitz is discussed. The explicit form of q-creation and q-annihilation operators, q-coherent states and an analog of the Fourier transformation are established. A connection of the kernel of this transform with a family of self-dual biorthogonal rational functions is observed.

The distribution of zeros of generalq-polynomials

“…Then In dealing with the q-harmonic oscillator, Askey and Suslov [16] have introduced the qpolynomials…”

“…continued fractions, Eulerian series, theta functions, elliptic functions, etc; see for instance [4,5]) and physics (e.g. angular momentum [6,7] and its q-analogue [8][9][10][11], the qSchrödinger equation [12] and q-harmonic oscillators [13][14][15][16][17][18][19]). Moreover, it is well known that the connection between the representation theory of quantum algebras (Clebsch-Gordan coefficients, 3j and 6j symbols) and the q-orthogonal polynomials (see [20,21,vol III,[22][23][24]), and the important role that these q-algebras play in physical applications (see for instance [26][27][28][29][30][31] and references therein).…”

An operator approach to the Al-Salam–Carlitz polynomials

“…To avoid this problem, we may restrict our attention to the case that v/u = q −r or y/x = q −r , where r is a nonnegative integer, as noticed by Askey and Suslov [4] and Fang [14].…”

“…The Al-Salam-Carlitz polynomials are q-orthogonal polynomials which arise in many applications such as the q-harmonic oscillator, theta functions, quantum groups and coding theory; see for example [4,5,15,16]. This paper presents an operator approach to the Rogers-type formulas and Mehler's formulas for the Al-Salam-Carlitz polynomials.…”

On a q-extension of the linear harmonic oscillator with the continuous orthogonality property on ℝ

“…Let us point out here that there are several publications (see [2]- [10] and references therein) devoted to the study of explicit realizations, which represent q-extensions of the Hermite functions (or the wave functions of the linear harmonic oscillator) H n (x) e −x 2 /2 . But none of these realizations satisfies all of the aforementioned requirements: the continuous weight functions in [2,4,7] are supported on the finite intervals; the continuous weight functions in [3,8] are not positive; the q-extensions in [2], [4]- [9] are not expressed in terms of polynomials in the independent variable; and, finally, the orthogonality relations in [5]- [7], [10] are discrete.…”