Properties of certain q-orthogonal polynomials are connected to the q-oscillator algebra. The Wall and q-Laguerre polynomials are shown to arise as matrix elements of q-exponentials of the generators in a representation of this algebra. A realization is presented where the continuous q-Hermite polynomials form a basis of the representation space. Various identities are interpreted within this model. In particular, the connection formula between the continuous big q-Hermite and the continuous q-Hermite polynomials is thus obtained, and two generating functions for these last polynomials are derived algebraically.

More on the q-oscillator algebra and q-orthogonal polynomials

Using the factorization method, we construct finite-difference Schrodinger operators (Jacobi matrices) whose discrete spectra are composed from independent arithmetic, or geometric series. Such systems originate from the periodic, orq-periodic closure of a chain of corresponding Darboux transformations. The Charlier, Krawtchouk, Meixner orthogonal polynomials, theirq-analogs, and some other classical polynomials appear as the simplest examples forN = 1 andN = 2 (N is the period of closure). A natural generalization involves discrete versions of the Painleve transcendents.

Difference schrödinger operators with linear and exponential discrete spectra

We study some of the algebraic structures that are compatible with the quantization of the harmonic oscillator through its Newton equation. Examples of such structures are given; they include undeformed and q-deformed oscillators, as well as the SU(2) and the deformed SUq(2) Lie algebras, which appear in a variety of physical models.

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Quantum algebraic structures compatible with the harmonic oscillator Newton equation

A realization of the Heisenberg q‐algebra whose generators are first‐order difference operators on the full real line is discussed herein. The eigenfunctions of the corresponding q‐oscillator Hamiltonian are given explicitly in terms of the q−1‐Hermite polynomials. The nonuniqueness of the measure for these q‐oscillator states is also studied.

https://www.fis.unam.mx/~bwolf/Articles/78.pdf

S. K. Suslov

Papers

A realization fo the q-harmonic oscillator