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 (formula presented) 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 < 0, that was actually encountered by the authors before in an application stemming from probability and complexity theory. We show that, in this case, the asymptotic expression and the expression for the upper bound differ only in a polynomial factor; whereas, the exponential factors are the same for both expressions. In addition, we present some numerical calculations using MAPLE (a computer program for performing symbolic and numerical computations), that show that both expressions are close to the actual value of the coefficients, even for moderate values of m.