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Factorization of the Riesz-Feller fractional quantum harmonic oscillators

Using the Riesz-Feller fractional derivative, we apply the factorization algorithm to the fractional quantum harmonic oscillator along the lines previously proposed by Olivar-Romero and Rosas-Ortiz, extending their results. We solve the non-Hermitian fractional eigenvalue problem in the k space by introducing in that space a new class of Hermite `polynomials' that we call Riesz-Feller Hermite `polynomials'. Using the inverse Fourier transform in Mathematica, interesting analytic results for the same eigenvalue problem in the x space are also obtained. Additionally, a more general factorization with two different Levy indices is briefly introduced