Q-exponential

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Definition

The q-exponential [e_q(z)] is defined as

[e_q(z)=\sum_^\infty \frac = \sum_^\infty \frac = \sum_^\infty z^n\frac) \cdots (1-q)}]

where [[n]_q!] is the q-factorial and

[(q;q)_n=(1-q^n)(1-q^)\cdots (1-q)]

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

[\left(\frac\right)_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\right)_q z^n = z^ \frac=[n]_q z^.]

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

For real [q>1], the function [e_q(z)] is an entire function of z. For [q<1], [e_q(z)] is regular in the disk [|z|<1/(1-q)].

For [q<1], 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) = \;_\phi_0 (0;q,z) = \prod_^\infty \frac ]

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