In
combinatorial
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ap ...
mathematics, a ''q''-exponential is a
''q''-analog of the
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
,
namely the
eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
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
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of variables with nonnegative integer expone ...
:
Here,
is the
''q''-bracket.
For other definitions of the ''q''-exponential function, see , , and .
Properties
For real
, the function
is an
entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...
of
. For
,
is regular in the disk
.
Note the inverse,
.
Addition Formula
The analogue of
does not hold for real numbers
and
. However, if these are operators satisfying the commutation relation
, then
holds true.
Relations
For