Q-exponential


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

Definition

The q-exponential is defined as
where 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, is the q-bracket.
For other definitions of the q-exponential function, see,, and.

Properties

For real, the function is an entire function of. For, is regular in the disk.
Note the inverse, .

Addition Formula

If, holds.

Relations

For , a function that is closely related is It is a special case of the basic hypergeometric series,
Clearly,

Relation with Dilogarithm

has the following infinite product representation:
On the other hand, holds.
When,
By taking the limit,
where is the dilogarithm.