q-exponential

The term q-exponential occurs in two contexts. The q-exponential distribution, based on the Tsallis q-exponential is discussed in elsewhere.

In combinatorial mathematics, a q-exponential is a q-analog of the exponential function,

namely the eigenfunction of a q-derivative. There are many q-derivatives, for example, the classical q-derivative, the Askey–Wilson operator, etc. Therefore, unlike the classical exponentials, q-exponentials are not unique. For example, $e_q(z)$ is the q-exponential corresponding to the classical q-derivative while $\mathcal{E}_q(z)$ are eigenfunctions of the Askey–Wilson operators.

The q-exponential is also known as the quantum dilogarithm.{{Cite web|last=Zudilin|first=Wadim|date=14 March 2006|title=Quantum dilogarithm|url=https://wain.mi.ras.ru/PS/mpim-mar2006.pdf|access-date=16 July 2021|website=wain.mi.ras.ru}}{{Cite journal|last1=Faddeev|first1=L.d.|last2=Kashaev|first2=R.m.|date=1994-02-20|title=Quantum dilogarithm|url=https://www.worldscientific.com/doi/abs/10.1142/S0217732394000447|journal=Modern Physics Letters A|volume=09|issue=5|pages=427–434|doi=10.1142/S0217732394000447|issn=0217-7323|arxiv=hep-th/9310070|bibcode=1994MPLA....9..427F|s2cid=119124642}}

Definition

The q-exponential $e_q(z)$ is defined as

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

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.

For other definitions of the q-exponential function, see {{harvtxt|Exton|1983}}, {{harvtxt|Ismail|Zhang|1994}}, and {{harvtxt|Cieśliński|2011}}.

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 $|z|<1/(1-q)$ .

Note the inverse, $~e_q(z) ~ e_{1/q} (-z) =1$ .

=Addition Formula=

The analogue of $\exp(x)\exp(y)=\exp(x+y)$ does not hold for real numbers $x$ and $y$ . However, if these are operators satisfying the commutation relation $xy=qyx$ , then $e_q(x)e_q(y)=e_q(x+y)$ holds true.{{cite book |last1=Kac |first1=V. |last2=Cheung |first2=P. |title=Quantum Calculus |date=2011 |publisher=Springer |isbn=978-1461300724 |page=31 |ref=KacCheung}}

Relations

For $-1, a function that is closely related is E_q(z). It is a special case of the basic hypergeometric series,$

: $E_{q}(z)=\;_{1}\phi_{1}\left({\scriptstyle{0\atop 0}}\, ;\,z\right)=\sum_{n=0}^{\infty}\frac{q^{\binom{n}{2}}(-z)^{n}}{(q;q)_{n}}=\prod_{n=0}^{\infty}(1-q^{n}z)=(z;q)_\infty.$

Clearly,

: $\lim_{q\to1}E_{q}\left(z(1-q)\right)=\lim_{q\to1}\sum_{n=0}^{\infty}\frac{q^{\binom{n}{2}}(1-q)^{n}}{(q;q)_{n}}
(-z)^{n}=e^{-z} .~$

=Relation with Dilogarithm=

$e_q(x)$ has the following infinite product representation:

: $e_q(x)=\left(\prod_{k=0}^\infty(1-q^k(1-q)x)\right)^{-1}.$

On the other hand, $\log(1-x)=-\sum_{n=1}^\infty\frac{x^n}{n}$ holds.

When $|q|<1$ ,

: $\begin{align}
\log e_q(x) &= -\sum_{k=0}^\infty\log(1-q^k(1-q)x) \\
&= \sum_{k=0}^\infty\sum_{n=1}^\infty\frac{(q^k(1-q)x)^n}{n} \\
&= \sum_{n=1}^\infty\frac{((1-q)x)^n}{(1-q^n)n} \\
&= \frac{1}{1-q}\sum_{n=1}^\infty\frac{((1-q)x)^n}{[n]_qn}
\end{align}.$

By taking the limit $q\to 1$ ,

: $\lim_{q\to 1}(1-q)\log e_q(x/(1-q))=\mathrm{Li}_2(x),$

where $\mathrm{Li}_2(x)$ is the dilogarithm.

References

{{cite journal

| last1=Cieśliński | first1=Jan L. | authorlink1=Jan L. Cieśliński

| date=2011

| title=Improved q-exponential and q-trigonometric functions

| journal=Applied Mathematics Letters

| volume=24

| issue=12

| pages=2110–2114

| doi=10.1016/j.aml.2011.06.009| s2cid=205496812 | doi-access=free

| arxiv=1006.5652

}}

{{cite book

| last1=Exton | first1=Harold | authorlink1=Harold Exton

| date=1983

| title=q-Hypergeometric Functions and Applications

| publisher=New York: Halstead Press, Chichester: Ellis Horwood

| isbn=0853124914}}

{{cite book

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| last2=Rahman | first2=Mizan Rahman | authorlink2=Mizan Rahman

| date=2004

| title=Basic Hypergeometric Series

| publisher=Cambridge University Press

| isbn=0521833574}}

{{cite book

| last1=Ismail | first1=Mourad E. H. | authorlink1=Mourad Ismail

| date=2005

| title=Classical and Quantum Orthogonal Polynomials in One Variable

| publisher=Cambridge University Press

| doi=10.1017/CBO9781107325982| isbn=9780521782012 }}

{{cite journal

| last1=Ismail | first1=Mourad E. H. | authorlink1=Mourad Ismail

| last2=Zhang | first2=Ruiming | authorlink2=Ruiming Zhang

| date=1994

| title=Diagonalization of certain integral operators

| journal=Advances in Mathematics

| volume=108

| issue=1

| pages=1–33

| doi=10.1006/aima.1994.1077 | doi-access=free}}

{{cite journal

| last1=Ismail | first1=Mourad E. H. | authorlink1=Mourad Ismail

| last2=Rahman | first2=Mizan | authorlink2=Mizan Rahman

| last3=Zhang | first3=Ruiming | authorlink3=Ruiming Zhang

| date=1996

| title=Diagonalization of certain integral operators II

| journal=Journal of Computational and Applied Mathematics

| volume=68

| issue=1–2

| pages=163–196

| doi=10.1016/0377-0427(95)00263-4 | doi-access=free| citeseerx=10.1.1.234.4251

}}

{{cite journal

| last1=Jackson | first1=F. H. | authorlink1=F. H. Jackson

| date=1909

| title=On q-functions and a certain difference operator

| journal=Transactions of the Royal Society of Edinburgh

| volume=46

| issue=2

| pages=253–281

| doi=10.1017/S0080456800002751| s2cid=123927312 }}

Category:Q-analogs

Category:Exponentials