Q-exponential

In combinatorial mathematics, the q-exponential is the q-analog of the exponential function.

Definition

The q-exponential $e\_q(z)$ is defined as:$e\_q(z)=sum\_\{n=0\}^infty\; frac\{z^n\}\{\; [n]\; \_q!\}\; =\; sum\_\{n=0\}^infty\; frac\{z^n\; (1-q)^n\}\{(q;q)\_n\}\; =\; sum\_\{n=0\}^infty\; z^nfrac\{(1-q)^n\}\{(1-q^n)(1-q^\{n-1\})\; cdots\; (1-q)\}$

where $[n]\; \_q!$ is the q-factorial and :$(q;q)\_n=(1-q^n)(1-q^\{n-1\})cdots\; (1-q)$

is the q-Pochhammer symbol. That this is the q-analog of the exponential follows from the property

:$left(frac\{d\}\{dz\}\; ight)\_q\; e\_q(z)\; =\; e\_q(z)$

where the derivative on the left is the q-derivative. The above is easily verified by considering the q-derivative of the monomial

:$left(frac\{d\}\{dz\}\; ight)\_q\; z^n\; =\; z^\{n-1\}\; frac\{1-q^n\}\{1-q\}=\; [n]\; \_q\; z^\{n-1\}.$

Here, $[n]\; \_q$ is the q-bracket.

Properties

For real $q>1$, the function $e\_q(z)$ is an entire function of "z". For , $e\_q(z)$ is regular in the disk .

Relations

For , a function that is closely related is

:$e\_q(z)\; =\; E\_q(z(1-q))$

Here, $E\_q(t)$ is a special case of the basic hypergeometric series:

:$E\_q(z)\; =\; ;\_\{1\}phi\_0\; (0;q,z)\; =\; prod\_\{n=0\}^infty\; frac\; \{1\}\{1-q^n\; z\}$