New $q-$Hermite polynomials: characterization, operator algebra and associated coherent states

TL;DR

This paper introduces new $q$-Hermite polynomials, fully characterizing their properties, operator algebra, and associated coherent states, advancing the mathematical framework of $q$-deformed oscillators.

Contribution

It provides a complete characterization of new $q$-Hermite polynomials, including recursive relations, differential equations, and the construction of related coherent states.

Findings

Derived explicit three-term recursive relation.

Established the second-order differential equation.

Constructed and analyzed associated coherent states.

Abstract

This paper addresses a construction of new $q-$Hermite polynomials with a full characterization of their main properties and corresponding raising and lowering operator algebra. The three-term recursive relation as well as the second-order differential equation obeyed by these new polynomials are explicitly derived. Relevant operator actions, including the eigenvalue problem of the deformed oscillator and the self-adjointness of the related position and momentum operators, are investigated and analyzed. The associated coherent states are constructed and discussed with an explicit resolution of the induced moment problem.