In mathematics, the quantum dilogarithm is a special function defined by the formula

: \phi(x)\equiv(x;q)\infty=\prod{n=0}^\infty (1-xq^n),\quad |q|

It is the same as the q-exponential function e_q(x).

Let u,v be "q-commuting variables", that is elements of a suitable noncommutative algebra satisfying Weyl's relation uv=qvu. Then, the quantum dilogarithm satisfies Schützenberger's identity :\phi(u) \phi(v)=\phi(u + v), Faddeev-Volkov's identity :\phi(v) \phi(u)=\phi(u +v -vu), and Faddeev-Kashaev's identity :\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 \Phi_b(w) is defined by the following formula:

: \Phi_b(z)=\exp \left( \frac{1}{4}\int_C \frac{e^{-2i zw }} {\sinh (wb) \sinh (w/b) } \frac{dw}{w} \right),

where the contour of integration 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:

: \Phi_b(x)=\exp\left(\frac{i}{2\pi}\int_{\mathbb R}\frac{\log(1+e^{tb^2+2\pi b x})}{1+e^{t}},dt\right).

Ludvig Faddeev discovered the quantum pentagon identity:

: \Phi_b(\hat p)\Phi_b(\hat q)

\Phi_b(\hat q) \Phi_b(\hat p+ \hat q) \Phi_b(\hat p), where \hat p and \hat q are self-adjoint (normalized) quantum mechanical momentum and position operators satisfying Heisenberg's commutation relation

:[\hat p,\hat q]=\frac1{2\pi i}

and the inversion relation

: \Phi_b(x)\Phi_b(-x)=\Phi_b(0)^2 e^{\pi ix^2},\quad \Phi_b(0)=e^{\frac{\pi i}{24}\left(b^2+b^{-2}\right)}.

The quantum dilogarithm finds applications in mathematical physics, quantum topology, cluster algebra theory.

The precise relationship between the q-exponential and \Phi_b is expressed by the equality

:\Phi_b(z)=\frac{E_{e^{2\pi ib^2}}(-e^{\pi ib^2+2\pi zb})}{E_{e^{-2\pi i/b^2}}(-e^{-\pi i/b^2+2\pi z/b})},

valid for \operatorname{Im} b^20.