Cubic decomposition of a family of semiclassical polynomial sequences of class one

Abstract

In the present work, we deal with a monic orthogonal polynomial sequence (MOPS) deduced by its cubic decomposition (CD) $W_{3n}(x)=P_n(x^3+qx+r)$. Our aim is to determine all semiclassical of class one of these sequences. From the CD structure, only three MOPS appear, which their recurrence coefficients are explicitly given. An integral representation of positive definite case is established.

Keywords:

• orthogonal polynomial sequences
• semiclassical forms
• cubic decomposition
• integral representation

Mathematics Subject Classification::

• Primary: 33C45
• Secondary: 42C05

Acknowledgments

The authors are very grateful to the referees for their constructive and valuable comments and for making us pay attention to a certain references. Their suggestions and remarks have contributed to improve substantially the presentation of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.