Some asymptotic results on q-binomial coefficients

TL;DR

This paper investigates the asymptotic behavior of coefficients in q-binomial coefficients with fixed k, providing new results involving Eulerian polynomials and their generalizations.

Contribution

It offers novel asymptotic results for q-binomial coefficients, connecting them with Eulerian polynomials and expanding understanding of their coefficient behavior.

Findings

Asymptotic formulas for fixed k

Connections with Eulerian polynomials

Generalizations of known results

Abstract

We look at the asymptotic behavior of the coefficients of the $q$-binomial coefficients (or Gaussian polynomials) $\binom{a+k}{k}_q$, when $k$ is fixed. We give a number of results in this direction, some of which involve Eulerian polynomials and their generalizations.