Some Theorems and Applications of the $(q,r)$-Whitney Numbers

TL;DR

This paper explores the properties, combinatorial interpretations, and applications of the $(q,r)$-Whitney numbers, including symmetric polynomial forms, convolution identities, and connections to discrete $q$-distributions.

Contribution

It introduces elementary and symmetric polynomial forms for the $(q,r)$-Whitney numbers and provides new combinatorial interpretations and applications in $q$-distributions.

Findings

Elementary and complete symmetric polynomial forms derived

Combinatorial interpretations via $A$-tableaux established

Applications to discrete $q$-distributions demonstrated

Abstract

The $(q,r)$-Whitney numbers were recently defined in terms of the $q$-Boson operators, and several combinatorial properties which appear to be $q$-analogues of similar properties were studied. In this paper, we obtain elementary and complete symmetric polynomial forms for the $(q,r)$-Whitney numbers, and give combinatorial interpretations in the context of $A$-tableaux. We also obtain convolution-type identities using the combinatorics of $A$-tableaux. Lastly, we present applications and theorems related to discrete $q$-distributions.