For certain nice rings, Bhargava constructs a generalized factorial via p-orderings; the details of which can be found in Bhargava. In example 18, of section 10, he states that the generalized factorial on the p-ordered set $\left\{\frac{q^n-1}{q-1}\right\}_{n \in\mathbb{N}} \subset \mathbb{C[q,q^{-1}]}$ recovers the q-factorial $\prod_{i=1}^n\frac{q^i-1}{q-1}$. When I use the method, I obtain $\prod_{k=0}^{n-1}\frac{q^n-q^k}{q-1}=q^{\frac{n(n-1)}{2}}\prod_{i=1}^{n}\frac{q^i-1}{q-1}.$
- Can someone explain why he does not have the extra factor of $q^{\frac{n(n-1)}{2}}$?
Perhaps, I didn't follow his procedure correctly..
$q$ in invertible in $\mathbb C[q,q^{-1}]$. Bhargava's factorial are defined as generators of the factorial ideals. These generators are defined up to an invertible.