Bounds of $q$—factorial $[n{]}_q!$

Abstract

In this paper, we obtain the following upper and lower bounds for $q$-factorial $[n{]}_q!$:

$$(q;q{)}_{\mathrm{\infty }}(1–q{)}^{-n}{e}^{f_q(n+1)}<[n{]}_q!<(q;q{)}_{\mathrm{\infty }}(1–q{)}^{-n}{e}^{g_q(n+1)},$$ where $n\ge 1$, $0<q<1$, 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 $g_q(n)$.