# Differences

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 — deformed_natural [2018/02/03 02:11] (current)nikolaj created 2018/02/03 02:11 nikolaj created 2018/02/03 02:11 nikolaj created Line 1: Line 1: + ===== 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] + ​ + + 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]] 