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deformed_natural [2018/02/03 02:11] (current)
nikolaj created
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 +===== 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]]
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