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In this work, the [Formula: see text]-Schrödinger equations with symmetric polynomial potentials are considered. The spectrum of the model is obtained for several values of [Formula: see text], and the limiting case as [Formula: see text] is considered. The Rayleigh–Ritz variational method is adopted to the system. The discrete [Formula: see text]-Hermite I polynomials are handled as basis in this method. Furthermore, the following potentials with numerous results are presented as applications: [Formula: see text]-harmonic, purely [Formula: see text]-quartic and [Formula: see text]-quartic oscillators. It is also shown that the obtained results confirm the ones that exist in the literature for the continuous case.

Discrete q-Hermite I polynomials are a member of the q-polynomials of the Hahn class. They are the polynomial solutions of a second order difference equation of hypergeometric type. These polynomials are one of the q-analogous of the Hermite polynomials. It is well known that the q-Hermite I polynomials approach the Hermite polynomials as q tends to 1. In this chapter, the orthogonality property of the discrete q-Hermite I polynomials is considered. Moreover, the orthogonality relation for the k-th order q-derivatives of the discrete q-Hermite I polynomials is obtained. Finally, it is shown that, under a suitable transformation, these relations give the corresponding relations for the Hermite polynomials in the limiting case as q goes to 1.

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