q-exponential

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

The q-exponential is also known as the quantum dilogarithm.

Definition

The q-exponential [math]\displaystyle{ e_q(z) }[/math] is defined as

[math]\displaystyle{ 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]\displaystyle{ [n]!_q }[/math] is the q-factorial and

[math]\displaystyle{ (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]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ [n]_q }[/math] is the q-bracket. For other definitions of the q-exponential function, see (Exton 1983), (Ismail Zhang), (Suslov 2003) and (Cieśliński 2011).

Properties

For real [math]\displaystyle{ q\gt 1 }[/math], the function [math]\displaystyle{ e_q(z) }[/math] is an entire function of [math]\displaystyle{ z }[/math]. For [math]\displaystyle{ q\lt 1 }[/math], [math]\displaystyle{ e_q(z) }[/math] is regular in the disk [math]\displaystyle{ |z|\lt 1/(1-q) }[/math].

Note the inverse, [math]\displaystyle{ ~e_q(z) ~ e_{1/q} (-z) =1 }[/math].

Addition Formula

The analogue of [math]\displaystyle{ \exp(x)\exp(y)=\exp(x+y) }[/math] does not hold for real numbers [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math]. However, if these are operators satisfying the commutation relation [math]\displaystyle{ xy=qyx }[/math], then [math]\displaystyle{ e_q(x)e_q(y)=e_q(x+y) }[/math] holds true.

Relations

For [math]\displaystyle{ -1\lt q\lt 1 }[/math], a function that is closely related is [math]\displaystyle{ E_q(z). }[/math] It is a special case of the basic hypergeometric series,

[math]\displaystyle{ 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]\displaystyle{ \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]\displaystyle{ e_q(x) }[/math] has the following infinite product representation:

[math]\displaystyle{ e_q(x)=\left(\prod_{k=0}^\infty(1-q^k(1-q)x)\right)^{-1}. }[/math]

On the other hand, [math]\displaystyle{ \log(1-x)=-\sum_{n=1}^\infty\frac{x^n}{n} }[/math] holds. When [math]\displaystyle{ |q|\lt 1 }[/math],

[math]\displaystyle{ \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]\displaystyle{ q\to 1 }[/math],

[math]\displaystyle{ \lim_{q\to 1}(1-q)\log e_q(x/(1-q))=\mathrm{Li}_2(x), }[/math]

where [math]\displaystyle{ \mathrm{Li}_2(x) }[/math] is the dilogarithm.

References

• "Improved q-exponential and q-trigonometric functions". Applied Mathematics Letters 24 (12): 2110–2114. 2011. doi:10.1016/j.aml.2011.06.009. 
• q-Hypergeometric Functions and Applications. New York: Halstead Press, Chichester: Ellis Horwood. 1983. ISBN 0853124914. 
• Basic Hypergeometric Series. Cambridge University Press. 2004. ISBN 0521833574. 
• Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge University Press. 2005. doi:10.1017/CBO9781107325982. ISBN 9780521782012. 
• "Diagonalization of certain integral operators". Advances in Mathematics 108 (1): 1–33. 1994. doi:10.1006/aima.1994.1077. 
• "Diagonalization of certain integral operators II". Journal of Computational and Applied Mathematics 68 (1–2): 163–196. 1996. doi:10.1016/0377-0427(95)00263-4. 
• "On q-functions and a certain difference operator". Transactions of the Royal Society of Edinburgh 46 (2): 253–281. 1909. doi:10.1017/S0080456800002751.