Multiple analogues of binomial coefficients and related families of special numbers

TL;DR

This paper introduces multidimensional generalizations of classical special numbers using $qt$-binomial coefficients, exploring their properties and establishing probability measures on integer partitions.

Contribution

It constructs multiple $qt$-analogues of well-known special numbers and investigates their fundamental properties and applications.

Findings

Defined new multidimensional $qt$-binomial coefficients

Established properties of $qt$-Stirling, Bell, Bernoulli, Catalan, and Fibonacci numbers

Proved probability measures on integer partitions

Abstract

We construct multiple $qt$-binomial coefficients and related multiple analogues of several celebrated families of special numbers in this paper. These multidimensional generalizations include the first and the second kind of $qt$-Stirling numbers, $qt$-Bell numbers, $qt$-Bernoulli numbers, $qt$-Catalan numbers and the $qt$--Fibonacci numbers. In the course of developing main properties of these extensions, we prove results that are significant in their own rights such as certain probability measures on the set of integer partitions.