Cariñena Orthogonal Polynomials Are Jacobi Polynomials

The relativistic Hermite polynomials (RHP) were introduced in 1991 by Aldaya et al. [3] in a generalization of the theory of the quantum harmonic oscillator to the relativistic context. These polynomials were later related to the more classical Gegenbauer (or more generally Jacobi) polynomials in a study by Nagel [4]. For this reason, they do not deserve any special study since their properties can be deduced from the properties of the well-known Jacobi polynomials a more general class of polynomials that includes Gegenbauer polynomials as underlined by Ismail in [6]. Recently, Cariñena et al. [2] studied an extension of the quantum harmonic oscillator on the sphere S2 and on the hyperbolic plane and they showed that the Schrdinger equation can be analytically solved; the solutions are an extension of the classical wavefunctions of the quantum harmonic oscillator where the usual Hermite polynomials are replaced by some new polynomials that we will call here Cariñena polynomials. In the next section, we show that these Cariñena polynomials are in fact Jacobi polynomials. The last section is devoted to the application of these results to the nonextensive context.