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} . Then, the quantum dilogarithm satisfies Schützenberger's identity
ϕ ( u ) ϕ ( v ) = ϕ ( u + v ) , {\displaystyle \phi (u)\phi (v)=\phi (u+v),}
Faddeev-Volkov's identity
ϕ ( v ) ϕ ( u ) = ϕ ( u + v − v u ) , {\displaystyle \phi (v)\phi (u)=\phi (u+v-vu),}
and Faddeev-Kashaev's identity
ϕ ( v ) ϕ ( u ) = ϕ ( u ) ϕ ( − v u ) ϕ ( v ) . {\displaystyle \phi (v)\phi (u)=\phi (u)\phi (-vu)\phi (v).}
The latter is known to be a quantum generalization of Rogers' five term dilogarithm identity. Faddeev's quantum dilogarithm Φ b ( w ) {\displaystyle \Phi _{b}(w)} is defined by the following formula:
Φ b ( z ) = exp ( 1 4 ∫ C e − 2 i z w sinh ( w b ) sinh ( w / b ) d w w ) , {\displaystyle \Phi _{b}(z)=\exp \left({\frac {1}{4}}\int _{C}{\frac {e^{-2izw}}{\sinh(wb)\sinh(w/b)}}{\frac {dw}{w}}\right),}
where the contour of integration C {\displaystyle C} goes along the real axis outside a small neighborhood of the origin and deviates into the upper half-plane near the origin. The same function can be described by the integral formula of Woronowicz:
Φ b ( x ) = exp ( i 2 π ∫ R log ( 1 + e t b 2 + 2 π b x ) 1 + e t d t ) . {\displaystyle \Phi _{b}(x)=\exp \left({\frac {i}{2\pi }}\int _{\mathbb {R} }{\frac {\log(1+e^{tb^{2}+2\pi bx})}{1+e^{t}}}\,dt\right).}
Ludvig Faddeev discovered the quantum pentagon identity:
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