Q-exponential

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title: "Q-exponential" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["q-analogs", "exponentials"] topic_path: "general/q-analogs" source: "https://en.wikipedia.org/wiki/Q-exponential" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

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.

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 , , and .

Properties

For real q1, the function e_q(z) is an entire function of z. For q, e_q(z) is regular in the disk |z|.

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.

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|,

:\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 | 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 }}

References

  1. Zudilin, Wadim. (14 March 2006). "Quantum dilogarithm".
  2. (1994-02-20). "Quantum dilogarithm". Modern Physics Letters A.
  3. (2011). "Quantum Calculus". Springer.

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