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 \(g_q(n)\).
Citation
M. Mansour, M.A. Obaid. Bounds of \(q\)—factorial \([n]_q!\)[J], Ars Combinatoria, Volume 102. 313-319. .