Hq-Semiclassical orthogonal polynomials via polynomial mappings

In this work we study orthogonal polynomials via polynomial mappings in the framework of the H-q-semiclassical class. We consider two monic orthogonal polynomial sequences {p(n)(x)}(n >= 0) and {qn(x)}(n >= 0) such that p(kn)(x) = q(n)(x(k)), n = 0, 1, 2,..., where k >= 2 is a fixed integer number, and we prove that if one of the sequences, {p(n)(x)}(n >= 0) or {q(n)(x)}(n >= 0), is H-q-semiclassical, then so is the other one. In particular, we show that if {p(n)(x)}(n >= 0) is H-q-semiclassical of class s <= k - 1, then {q(n)(x)}(n >= 0) is H-qk-classical. This fact allows us to recover and extend recent results in the framework of cubic transformations (k = 3). We also provide illustrative examples of H-q -semiclassical sequences of classes 1 and 2 involving little q-Laguerre and little q-Jacobi polynomials, including discrete measure representations for some of the considered examples.