$H_q-$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 $\{p_n(x){\}}_{n\ge 0}$ and $\{q_n(x){\}}_{n\ge 0}$ such that $$p_{kn}(x)=q_n(x^k),n=0,1,2,\dots ,$$ being $k$ a fixed integer number such that $k\ge 2$, and we prove that if one of the sequences $\{p_n(x){\}}_{n\ge 0}$ or $\{q_n(x){\}}_{n\ge 0}$ is $H_q-$semiclassical, then so is the other one. In particular, we show that if $\{p_n(x){\}}_{n\ge 0}$ is $H_q-$semiclassical of class $s\le k-1$, then $\{q_n(x){\}}_{n\ge 0}$ 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.