$H_q-$semiclassical orthogonal polynomials via polynomial mappings

Abstract: 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 ${p_n (x)}{n\geq0}$ and ${q_n(x)}{n\geq0}$ such that $$ p_{kn}(x)=q_n(x^k)\;,\quad n=0,1,2,\ldots\;, $$ being $k$ a fixed integer number such that $k\geq2$, and we prove that if one of the sequences ${p_n (x)}{n\geq0}$ or ${q_n(x)}{n\geq0}$ is $H_q-$semiclassical, then so is the other one. In particular, we show that if ${p_n(x)}{n\geq0}$ is $H_q-$semiclassical of class $s\leq k-1$, then ${q_n (x)}{n\geq0}$ is $H_{q^k}-$classical. This fact allows us to recover and extend recent results in the framework of cubic transformations, whenever we consider the above equality with $k=3$. The idea of blocks of recurrence relations introduced by Charris and Ismail plays a key role in our study.