Information About

Q-exponential

DEFINITION

The q-exponential

e_q(z)

is defined as
:

e_q(z)=
\sum_{n=0}^\infty rac{z^n}{

{Link without Title} _q!} = \sum_{n=0}^\infty rac{z^n (1-q)^n}{(q;q)_n} = \sum_{n=0}^\infty z^nrac{(1-q)^n}{(1-q^n)(1-q^{n-1}) \cdots (1-q)}

where

{Link without Title} _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(rac{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(rac{d}{dz}
ight)_q z^n = z^{n-1} rac{1-q^n}{1-q}
=

{Link without Title} _q z^{n-1}.

Here,

{Link without Title} _q is the Q-bracket .

PROPERTIES