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, <math>e_q(z)</math> is the ''q''-exponential corresponding to the classical ''q''-derivative while <math>\mathcal{E}_q(z)</math> are eigenfunctions of the Askey–Wilson operators.

The ''q''-exponential is also known as the quantum dilogarithm.<ref>{{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|archive-date=5 July 2017|archive-url=https://web.archive.org/web/20170705105354/http://wain.mi.ras.ru/PS/mpim-mar2006.pdf|url-status=dead}}</ref><ref>{{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}}</ref>

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

==Properties== For real <math>q>1</math>, the function <math>e_q(z)</math> is an entire function of <math>z</math>. 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>.

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

==Relations== For <math>-1<q<1</math>, a function that is closely related is <math>E_q(z).</math> It is a special case of the basic hypergeometric series,

:<math>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. </math>

Clearly, :<math>\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} .~ </math>

===Relation with Dilogarithm=== <math>e_q(x)</math> has the following infinite product representation: :<math>e_q(x)=\left(\prod_{k=0}^\infty(1-q^k(1-q)x)\right)^{-1}. </math> On the other hand, <math>\log(1-x)=-\sum_{n=1}^\infty\frac{x^n}{n}</math> holds. When <math>|q|<1</math>,

:<math>\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}.</math>

By taking the limit <math>q\to 1</math>, :<math>\lim_{q\to 1}(1-q)\log e_q(x/(1-q))=\mathrm{Li}_2(x), </math> where <math>\mathrm{Li}_2(x)</math> is the dilogarithm.

==References== {{Reflist}} * {{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 | last1=Gasper | first1=George | authorlink1=George Gasper | 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 }}

{{DEFAULTSORT:Q-Exponential}} Category:Q-analogs Category:Exponentials