The term q-exponential occurs in two contexts. The q-exponential distribution, based on the Tsallis q-exponential is discussed in elsewhere.

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 is the q-exponential corresponding to the classical q-derivative are eigenfunctions of the Askey–Wilson operators.

The q-exponential is also known as the quantum dilogarithm.

Definition

The q is defined as

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

is the q-bracket. For other definitions of the q-exponential function, see , , and .

Properties

is an entire function.

.

Addition Formula

holds true.

Relations

It is a special case of the basic hypergeometric series,

Clearly,

Relation with Dilogarithm

has the following infinite product representation:

,

,

is the dilogarithm.

References