In the fields of representation theory and quantum algebra, we often study quantized versions of algebraic objects by regarding them as algebras over $\mathbf{C}(q)$, or some subring of $\mathbf{C}(q)$, and using that algebra structure to twist the multiplication. Now I often see the introduction of a different indeterminate $v$ such that $v^2=q$, and we work over $\mathbf{C}(v)$ instead. Furthermore, I usually see a different version of the $q$-analogue of $n$ being used in this context. Letting $n_q$ be the usual $q$-analogue of the integer $n$ as defined in Quantum Calculus by Kac and Cheung, I often seen this different analogue $n_v$, where $$ n_q \;=\; \frac{q^n-1}{q-1} \qquad\qquad n_v \;=\; q^{\frac{1-n}{2}}n_q \;=\; \frac{v^n-v^{-n}}{v-v^{-1}} $$
What is the motivation of considering $v = q^\frac{1}{2}$ instead of $q$, and considering this scaled $q$-analogue $n_v$?
The most obvious motive for $n_v$ is its satisfying $n_{1/v}=n_v=\sqrt{n_qn_{1/q}}$. By contrast, $n_{1/q}=q^{1-n}\frac{q^n-1}{q-1}$.