Factorization of the hypergeometric-type difference equation on the non-uniform lattices: dynamical algebra

TL;DR

This paper demonstrates how to factorize hypergeometric-type difference equations on non-uniform lattices, revealing a connection to the dynamical algebra $su_q(1,1)$ and unifying models with $q$-linear spectra.

Contribution

It introduces a general factorization method for hypergeometric difference equations on non-uniform lattices, linking them to the $su_q(1,1)$ algebra and unifying various models.

Findings

Factorization of difference equations on non-uniform lattices is possible.

Most cases with $q$-linear spectra lead to the $su_q(1,1)$ algebra.

Explicit construction of algebra generators from difference operators.

Abstract

We argue that one can factorize the difference equation of hypergeometric type on the nonuniform lattices in general case. It is shown that in the most cases of q-linear spectrum of the eigenvalues this directly leads to the dynamical symmetry algebra $su_q(1,1)$, whose generators are explicitly constructed in terms of the difference operators, obtained in the process of factorization. Thus all models with the $q$-linear spectrum (some of them, but not all, previously considered in a number of publications) can be treated in a unified form.