On the nested sums and orthogonal polynomials

Abstract

The nested sums applied to a general three-term recurrence relation permits us to give compact representations of orthogonal polynomial sequences, which satisfy such kind of linear recurrence. We illustrate this model on particular examples of classical as well as nonclassical orthogonal polynomials. Some related nontrivial combinatorial identities are also obtained.

Keywords:

• combinatorial identities
• compact representations
• difference equations
• nested sums
• orthogonal polynomials
• three-term recurrence relation

AMS Subject Classifications::

• 05A19
• 05E35
• 11B83
• 15A15
• 42C05

Acknowledgements

The authors express their sincere thanks to the reviewers for their constructive comments and remarks in the improvement of this article.