Template:Distinguish2
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
is defined as
![{\displaystyle 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)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f224e5c169bd7d785bc27d1cb5c515c5416330d1)
where
is the q-factorial and
![{\displaystyle (q;q)_{n}=(1-q^{n})(1-q^{n-1})\cdots (1-q)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/352c9ce4fb89dad19bf549784ea3e9e2c0a6158f)
is the q-Pochhammer symbol. That this is the q-analog of the exponential follows from the property
![{\displaystyle \left({\frac {d}{dz}}\right)_{q}e_{q}(z)=e_{q}(z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11dad1ef314348eda0c93effeaf6a67d529b0303)
where the derivative on the left is the q-derivative. The above is easily verified by considering the q-derivative of the monomial
![{\displaystyle \left({\frac {d}{dz}}\right)_{q}z^{n}=z^{n-1}{\frac {1-q^{n}}{1-q}}=[n]_{q}z^{n-1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5454c7904c36ab46a23fd93c086675aa4f7f618)
Here,
is the q-bracket.
Properties
For real
, the function
is an entire function of z. For
,
is regular in the disk
.
Note the inverse,
.
Relations
For
, a function that is closely related is
![{\displaystyle e_{q}(z)=E_{q}(z(1-q)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c991c364f53f1b4688fbc24daa13d992328a3cf3)
Here,
is a special case of the basic hypergeometric series:
![{\displaystyle E_{q}(z)=\;_{1}\phi _{0}(0;q,z)=\prod _{n=0}^{\infty }{\frac {1}{1-q^{n}z}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bff81a709c7bd51782fe283e9706df665ad9115)
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