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.