Upper bounds and asymptotics for the q-binomial coefficients

In this article, we a derive an upper bound and an asymptotic formula for t he q-binomial, or Gaussian, coefficients. The q-binomial coefficients, that are defined by the expression ((m)(n))(q) = (1 - q(m))(1 - q(m-1))(...)(1 - q(m-n+1))/(1 - q(n))(1 - q(n- 1))(...)(1- q) 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 me ans. 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 asympto tic expression and the expression for the upper bound differ only in a poly nomial factor; whereas, the exponential factors are the same for both expre ssions. 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 coefficient s, even for moderate values of m.