Hq-Semiclassical orthogonal polynomials via polynomial mappings

In this work we study orthogonal polynomials via polynomial mappings in the framework of the Hq-semiclassical class. We consider two monic orthogonal polynomial sequences {pn(x)}n≥0 and {qn(x)}n≥0 such that pkn(x)=qn(xk),n=0,1,2,…,
where k≥2 is a fixed integer number, and we prove that if one of the sequences, {pn(x)}n≥0 or {qn(x)}n≥0, is Hq-semiclassical, then so is the other one. In particular, we show that if {pn(x)}n≥0 is Hq-semiclassical of class s≤k−1, then {qn(x)}n≥0 is Hqk-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 Hq-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.

orthogonal polynomials; q-polynomials; hq-semiclassical orthogonal polynomials; polynomial mappings; q-difference equations

Hq-Semiclassical orthogonal polynomials via polynomial mappings Articles