In combinatorial mathematics, a q-exponential is a q-analog of the exponential function, namely the eigenfunction of a q-derivative. There are many q-derivatives, for example, the classical q-derivative, the Askey–Wilson operator, etc. Therefore, unlike the classical exponentials, q-exponentials are not unique. For example, e_q(z) is the q-exponential corresponding to the classical q-derivative while are eigenfunctions of the Askey–Wilson operators. The q-exponential is also known as the quantum dilogarithm.

Definition

The q-exponential e_q(z) is defined as where [n]!_q is the q-factorial and is the q-Pochhammer symbol. That this is the q-analog of the exponential follows from the property where the derivative on the left is the q-derivative. The above is easily verified by considering the q-derivative of the monomial Here, [n]_q is the q-bracket. For other definitions of the q-exponential function, see, , and.

Properties

For real q>1, the function e_q(z) is an entire function of z. For q<1, e_q(z) is regular in the disk |z|<1/(1-q). Note the inverse,.

Addition Formula

The analogue of does not hold for real numbers x and y. However, if these are operators satisfying the commutation relation xy=qyx, then holds true.

Relations

For -1<q<1, a function that is closely related is E_q(z). It is a special case of the basic hypergeometric series, Clearly,

Relation with Dilogarithm

e_q(x) has the following infinite product representation: On the other hand, holds. When |q|<1, By taking the limit q\to 1, where is the dilogarithm.