===== Deformed natural =====
==== Function ====
| @#55CCEE: context     | @#55CCEE: $p\in Q$ |
| @#55CCEE: context     | @#55CCEE: $u:{\mathbb N}\to{}Q\to{\mathbb A}$ |
| @#FF9944: definition  | @#FF9944: $[n]_u(q) := \sum_{k=1}^n \dfrac{u_k(q)}{u_k(p)}$ |

And clearly the denominator must be nonzero.

=== Discussion ===
E.g., for another sequence $a_n$ consider $u(n,q):=q^{a_n}$.

In particular, consider $a_n:=n*x+d$ for some $d$. 

<code>
a[k_] = k x + d;
Sum[q^a[k], {k, 1, n}]
Limit[%, q -> 1]
</code>

In particular, consider $x:=1, d:=0$ for [[quantum_integer]]s.

=== Theorems ===
$\lim_{q\to p}[n]_u(q) = \sum_{k=1}^n 1 = n$

Also, for any sequence $(a_k)$,

$\sum_{k=1}^n a_k = \lim_{q\to 1}\dfrac{\partial}{\partial{}q} \sum_{k=1}^n q^{a_k}$

=== Reference ===
Wikipedia: [[http://en.wikipedia.org/wiki/Q-analog|q-analog]]

-----
=== Requirements ===
[[Metric space]]
=== Related ===
[[quantum_integer]]