Quantum Calculus
Table of Contents
- Calculus without limits
1. Quantum Derivative
1.1. Definition
\[ D_qf(x)=\frac{f(xq)-f(x)}{xq-x} \] where \(q\) is a formal variable.
It is identical to regular derivative when \(q\rightarrow1\) . Try substituting \(q=\frac{x-h}{x}\) .
2. q-Analog of Integer
2.1. Definition
\[ [n]_q = \frac{1-q^n}{1-q}=\sum_{k=0}^{n-1}q^k \]
3. Quantum Function
In power series of analytic functions, substitute every integer with q-analog of that integer.
4. q-Analog of Factorial
4.1. Definition
\[ [n]!_q=\prod_{k=1}^n[k]_q=\prod_{k=1}^n\frac{1-q^{k}}{1-q} \]
4.2. Application
It is a generating function for the number of permutation with \(n\) inversions. Proof of this involves an equation for the generating function. \[ [n+1]!_q=(1+q+\cdots+q^n)[n]!_q \] with each \(q^k\) corresponds to inserting \(n+1\) th element into \(n+1-k\) th position, since multiplying by \(q^k\) is to add \(k\) inversions.