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.

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.

[1][2] 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 Exton (1983), Ismail & Zhang (1994), and Cieśliński (2011).

, the function

is an entire function of

is regular in the disk

Note the inverse,

The analogue of

exp ⁡ ( x ) exp ⁡ ( y ) = exp ⁡ ( x + y )

does not hold for real numbers

However, if these are operators satisfying the commutation relation

holds true.

, a function that is closely related is

It is a special case of the basic hypergeometric series, Clearly,

has the following infinite product representation: On the other hand,

, By taking the limit

is the dilogarithm.