Upper Bounds and Asymptotics for the q-Binomial Coefficients

In this article, we a derive an upper bound and an asymptotic formula for the q‐binomial, or Gaussian, coefficients. The q‐binomial coefficients, that are defined by the expression are a generalization of the binomial coefficients, to which they reduce as q tends toward 1. In this article, we give an expression that captures the asymptotic behavior of these coefficients using the saddle point method and compare it with an upper bound for them that we derive using elementary means. We then consider as a case study the case q=1+z/m, z