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

where \(k\ge 2\) is a fixed integer number, 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 (\(k=3\)). 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.

在这项工作中，我们通过\（H_q \）-半经典类的框架中的多项式映射研究正交多项式。我们考虑两种首一正交多项式序列\（\ {P_N（X）\} _ {N \ GE 0} \）和\（\ {Q_N（X）\} _ {N \ GE 0} \） ，使得

其中\（k \ ge 2 \）是一个固定的整数，我们证明了如果其中一个序列是\（\ {p_n（x）\} _ {n \ ge 0} \）或\（\ {q_n （x）\} _ {n \ ge 0} \），是\（H_q \）-半经典的，那么另一个也是。特别地，我们证明如果\（\ {p_n（x）\} _ {n \ ge 0} \）是\（H_q \） -类\（s \ le k-1 \）的半经典，则\（ \ {q_n（x）\} _ {n \ ge 0} \）是\（H_ {q ^ k} \）-经典。这个事实使我们能够在三次变换（\（k = 3 \））的框架内恢复和扩展最近的结果。我们还提供\（H_ {q} \）的说明性示例-涉及很少q -Laguerre和很少q -Jacobi多项式的类1和2的半经典序列，包括一些所考虑示例的离散测度表示。