Deformed natural

Function

context $p\in Q$
context $u:{\mathbb N}\to{}Q\to{\mathbb A}$
definition $[n]_u(q) := \sum_{k=1}^n \dfrac{u_k(q)}{u_k(p)}$

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 $d$.

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

In particular, consider $x:=1, d:=0$ for quantum_integers.

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: q-analog


Requirements

Metric space

quantum_integer