It seems that most of the literature dealing with q-analogs defines q-numbers according to $[n]_q\equiv \frac{q^n-1}{q-1}.$ Even Mathematica uses this definition: with the built-in function QGamma you obtain
QGamma[n+1,q] / QGamma[n,q] = (q^n-1) / (q-1)
However, in some books and papers you find that the authors use a different notion of q-numbers, namely $[n]_q\equiv \frac{q^n-q^{-n}}{q-q^{-1}}.$ To me it seems that this alternative definition is exclusively used in the context of physics dealing with quantum groups.
Now my questions:
What is the notion/motivation for the latter definition? Somehow, it seems to be more relevant in (physical) applications.
Are these two approaches equivalent? If so, to what extent?
On the technical side: The QGamma function can be defined in terms of QPochhammer functions (see e.g. http://en.wikipedia.org/wiki/Q-gamma_function ). If I wanted to define a QGamma function such that $\frac{\Gamma_q(n+1)}{\Gamma_q(n)} = \frac{q^n-q^{-n}}{q-q^{-1}}$ what would be the (modified) definition of the respective QPochhammer functions?
Thanks in advance to everyone considering my questions!
P.S.: I'd also appreciate, if anyone can suggest some good literature on this topic in general.