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

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

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 $\left|z\right|<1/(1-q)$.

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:

$E_q(z)={}_1{\varphi }_0(0;q,z)=\prod _{n=0}^{\mathrm{\infty }}\frac{1}{1-q^{n}z}.$

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