A Recurrence Relation for Multiple Catalan Numbers

Document Type


Degree Name

Master of Science (MS)



Date of Award

Spring 2016


The goal of this project was to find a recurrence formula for certain q and qt analogs of classical Catalan numbers, and study such formulas in detail in the two dimensional case. To this end, we reviewed definitions and fundamental properties of classical Catalan numbers, q-Catalan numbers, and qt-Catalan numbers such as their generating functions and recursive relations. We found four recurrence relations in the literature for the classical Catalan numbers. We were able to generalized one of them to the q case and the multiple qt case. In fact, we wrote the recurrence relation in two different ways: in terms of rational Macdonald functions, and in combinatorial form in terms of Young tableau. Moreover, we worked simplified versions of both these n dimensional formulas in the two dimensional case to investigate possible connections to the classical case. This project covers one aspect of a new line of research in mathematics, and generalizes useful identities to higher dimensions.


Hasan Coskun

Subject Categories

Mathematics | Physical Sciences and Mathematics