Quantum dilogarithm
In mathematics, the quantum dilogarithm is a special function defined by the formula ϕ ( x ) ≡ ( x ; q ) ∞ = ∏ n = 0 ∞ ( 1 − x q n ) , | q | < 1 {\displaystyle \phi (x)\equiv (x;q)_{\infty }=\prod _{n=0}^{\infty }(1-xq^{n}),\quad |q|<1} It is the same as the q-exponential function e q ( x ) {\displaystyle e_{q}(x)} . Let u , v {\displaystyle u,v} be "q-commuting variables", that is elements of a suitable noncommutative algebra satisfying Weyl's relation u v = q v u {\displaystyle uv=qvu} .