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 is defined as
where is the q-factorial and
is the q-Pochhammer symbol. That this is the q-analog of the exponential follows from the property
where the derivative on the left is the q-derivative. The above is easily verified by considering the q-derivative of the monomial
Here, is the q-bracket.
Properties
For real , the function is an entire function of z. For , is regular in the disk .
Relations
For , a function that is closely related is
Here, is a special case of the basic hypergeometric series:
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