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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: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 eq(z) is defined as

eq(z)=n=0zn[n]q!=n=0zn(1q)n(q;q)n=n=0zn(1q)n(1qn)(1qn1)(1q)

where [n]q! is the q-factorial and

(q;q)n=(1qn)(1qn1)(1q)

is the q-Pochhammer symbol. That this is the q-analog of the exponential follows from the property

(ddz)qeq(z)=eq(z)

where the derivative on the left is the q-derivative. The above is easily verified by considering the q-derivative of the monomial

(ddz)qzn=zn11qn1q=[n]qzn1.

Here, [n]q is the q-bracket.

Properties

For real q>1, the function eq(z) is an entire function of z. For q<1, eq(z) is regular in the disk |z|<1/(1q).

Note the inverse, eq(z)e1/q(z)=1.

Relations

For q<1, a function that is closely related is

eq(z)=Eq(z(1q)).

Here, Eq(t) is a special case of the basic hypergeometric series:

Eq(z)=1ϕ0(0;q,z)=n=011qnz.

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