Schottky defect

Template:LowercaseIn combinatorial mathematics, the q-exponential is a q-analog of the exponential function, namely the eigenfunction of the q-derivative

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 $q<1$, 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:

$E_{q}(z)=\;_{1}\phi _{0}(0;q,z)=\prod _{n=0}^{\infty }{\frac {1}{1-q^{n}z}}~.$

References

• F. H. Jackson (1908), "On q-functions and a certain difference operator", Trans. Roy. Soc. Edin., 46 253-281.

• Exton, H. (1983), q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis Horwood, 1983, ISBN 0853124914, ISBN 0470274530, ISBN 978-0470274538

• Gasper G., and Rahman, M. (2004), Basic Hypergeometric Series, Cambridge University Press, 2004, ISBN 0521833574