Orthogonal polynomials and geometry of the quantum harmonic oscillator on constant curvature surfaces

In this talk, starting from results by Carinena et al. [Ann. Phys. 322, 434, 2007] about the quantum harmonic oscillator on constant (positive or negative) curvature surfaces, I will show some properties of the orthogonal polynomials associated with the corresponding wavefunctions. These polynomials have a strong connection with the hyperspherical polynomials, from which they inherit some properties. Moreover, a geometric transformation between the cases of a positive and a negative curvature surface can be made explicit: this transformation can be given an algebraic interpretation in terms of these orthogonal polynomials. Finally, a link can be exhibited with the canonical probability measures involved in the theory of nonextensive statistics.