Quadratic decomposition of a family ofHq-semiclassical orthogonal polynomial sequences

We deal with monic orthogonal polynomial sequences (MOPSs), , satisfying the three-term recurrence relation , , with initial conditions and , where and for all . These sequences are characterized by the relation , where is a MOPS. In this paper, we show that the sequence is -semiclassical if and only if the sequence is -semiclassical. Then, we express the characteristic elements of the -semiclassical sequence , such as the q-Pearson equation satisfied by the corresponding linear functional, the class of the linear functional, the first-order linear q-difference equation satisfied by the Stieltjes function and the coefficients of the structure relation for such a sequence of polynomials, in terms of the characteristic elements of the sequence, . In particular, if the sequence is -semiclassical of class zero, then we obtain a new non-symmetric -semiclassical sequence of polynomials of class s = 1.