Q-exponential

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

[edit] Definition

The q-exponential eq(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^n\frac{(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-series. That this is the q-analog of the exponential follows from the property

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

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

[edit] Properties

For real q > 1, the function eq(z) is an entire function of z. For q < 1, eq(z) is regular in the disk | z | < 1 / (1 − q).

[edit] Relations

For q < 1, a function that is closely related is

eq(z) = Eq(z(1 − q))

Here, Eq(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}
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