q-exponential

In combinatorial mathematics, a q-exponential is a q-analog of the exponential function, namely the eigenfunction 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, $e_q(z)$ is the q-exponential corresponding to the classical q-derivative while $\mathcal{E}_q(z)$ are eigenfunctions of the Askey-Wilson operators.

Definition

The q-exponential $e_q(z)$ is defined as

$e_q(z)= \sum_{n=0}^\infty \frac{z^n}{[n]_q!} = \sum_{n=0}^\infty \frac{z^n (1-q)^n}{(q;q)_n} = \sum_{n=0}^\infty z^n\frac{(1-q)^n}{(1-q^n)(1-q^{n-1}) \cdots (1-q)}$

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

$(q;q)_n=(1-q^n)(1-q^{n-1})\cdots (1-q)$

is the q-Pochhammer symbol. 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.

Properties

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 $|z|<1/(1-q)$.

Note the inverse, $~e_q(z) ~ e_{1/q} (-z) =1$.

Relations

For $-1<q<1$, a function that is closely related is $E_q(z).$ It is a special case of the basic hypergeometric series,

$E_{q}(z)=\;_{0}\phi_{1}(-;0,-z)=\sum_{n=0}^{\infty}\frac{q^{\binom{n}{2}}z^{n}}{(q;q)_{n}}=\prod_{n=0}^{\infty}(1-q^{n}z) ~.$

Clearly,

$\lim_{q\to1}E_{q}\left(z(1-q)\right)=\lim_{q\to1}\sum_{n=0}^{\infty}\frac{q^{\binom{n}{2}}(1-q)^{n}}{(q;q)_{n}}z^{n}=e^{z} .~$

References

• Exton, H. (1983), q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis Horwood, ISBN 0853124914, ISBN 0470274530, ISBN 978-0470274538
• Gasper, G. & Rahman, M. (2004), Basic Hypergeometric Series, Cambridge University Press, ISBN 0521833574
• Ismail, M. E. H. (2005), Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press.
• Jackson, F. H. (1908), "On q-functions and a certain difference operator", Transactions of the Royal Society of Edinburgh, 46, 253-281.