( ℛ , p , q ) $$( \mathcal {R}, p,q)$$ -Rogers–Szegö and Hermite Polynomials, and Induced Deformed Quantum Algebras

Abstract

AbstractDeformed quantum algebras, namely the q-deformed algebras and their extensions to (p, q)-deformed algebras, continue to attract much attention. One of the main reasons is that these topics represent a meeting point of nowadays fast developing areas in mathematics and physics like the theory of quantum orthogonal polynomials and special functions, quantum groups, integrable systems, quantum and conformal field theories and statistics.This contribution paper aims at characterizing the \(({\mathcal R},p,q)\)-Rogers–Szegö polynomials, and the \(({\mathcal R},p,q)\)-deformed difference equation giving rise to raising and lowering operators. These polynomials induce some realizations of generalized deformed quantum algebras, (called \(({\mathcal R},p,q)\)-deformed quantum algebras), which are here explicitly constructed. The study of continuous \(({\mathcal R},p,q)\)-Hermite polynomials is also performed. Known particular cases are recovered.KeywordsQuantum algebras \(({\mathcal R}{,\,}p{,\,}q)\)-deformed quantum algebrasOrthogonal polynomialsRogers–Szegö polynomialsHermite polynomialsMathematics Subject Classification (2000)Primary 33C45; Secondary 20G42