2017/12/18 by Castillo, K., De Jesus, M. N., Marcellán, F. +1
#33C45 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.1712.06474
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,… , being k a fixed integer number such that k≥2, 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, 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.