$H_q-$semiclassical orthogonal polynomials via polynomial mappings

TL;DR

This paper explores the relationship between polynomial mappings and $H_q$-semiclassical orthogonal polynomials, establishing conditions under which the semiclassical property is preserved or transformed, especially focusing on cubic transformations.

Contribution

It proves that polynomial mappings preserve $H_q$-semiclassical properties and extends recent results on cubic transformations, connecting classes of orthogonal polynomials through these mappings.

Findings

If one polynomial sequence is $H_q$-semiclassical, the related sequence via polynomial mapping is also $H_q$-semiclassical.

For class $s \\leq k-1$, the mapped sequence becomes $H_{q^k}$-classical.

The results recover and extend recent findings in cubic transformations.

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.