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{{Distinguish2|the Tsallis [[Tsallis statistics#q-exponential|q-exponential]]}} | |||
{{Lowercase}}In [[combinatorics|combinatorial]] [[mathematics]], the '''q-exponential''' is a [[q-analog]] of the [[exponential function]], | |||
namely the eigenfunction of the [[q-derivative]] | |||
==Definition== | |||
The q-exponential <math>e_q(z)</math> is defined as | |||
:<math>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)}</math> | |||
where <math>[n]_q!</math> is the [[q-factorial]] and | |||
:<math>(q;q)_n=(1-q^n)(1-q^{n-1})\cdots (1-q)</math> | |||
is the [[q-Pochhammer symbol]]. That this is the q-analog of the exponential follows from the property | |||
:<math>\left(\frac{d}{dz}\right)_q e_q(z) = e_q(z)</math> | |||
where the derivative on the left is the [[q-derivative]]. The above is easily verified by considering the q-derivative of the [[monomial]] | |||
:<math>\left(\frac{d}{dz}\right)_q z^n = z^{n-1} \frac{1-q^n}{1-q} | |||
=[n]_q z^{n-1}.</math> | |||
Here, <math>[n]_q</math> is the [[q-bracket]]. | |||
==Properties== | |||
For real <math>q>1</math>, the function <math>e_q(z)</math> is an [[entire function]] of ''z''. For <math>q<1</math>, <math>e_q(z)</math> is regular in the disk <math>|z|<1/(1-q)</math>. | |||
Note the inverse, <math>~e_q(z) ~ e_{1/q} (-z) =1</math>. | |||
==Relations== | |||
For <math>q<1</math>, a function that is closely related is | |||
:<math>e_q(z) = E_q(z(1-q)).</math> | |||
Here, <math>E_q(t)</math> is a special case of the [[basic hypergeometric series]]: | |||
:<math>E_q(z) = \;_{1}\phi_0 (0;q,z) = \prod_{n=0}^\infty | |||
\frac {1}{1-q^n z} ~. </math> | |||
==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 | |||
{{DEFAULTSORT:Q-Exponential}} | |||
[[Category:Q-analogs]] | |||
[[Category:Exponentials]] |
Revision as of 03:34, 4 February 2014
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