Bounds of \(q\)—factorial \([n]_q!\)

M. Mansour1, M.A. Obaid1
1King AbdulAziz University, Faculty of Science, Mathematics Department, P, 0. Box 80203, Jeddah 21589 , Saudi Arabia.

Abstract

In this paper, we obtain the following upper and lower bounds for \(q\)-factorial \([n]_q!\):

\[(q; q)_\infty (1 – q)^{-n} e^{f_q(n+1)} < [n]_q! < (q; q)_\infty (1 – q)^{-n} e^{g_q(n+1)},\] where \(n \geq 1\), \(0 < q < 1\), and the two sequences \(f_q(n)\) and \(g_q(n)\) tend to zero through positive values. Also, we present two examples of the two sequences \(f_q(n)\) and \(g_q(n)\).