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, ${\displaystyle e_{q}(z)}$ is the q-exponential corresponding to the classical q-derivative while ${\displaystyle {\mathcal {E}}_{q}(z)}$ are eigenfunctions of the Askey-Wilson operators.

Not to be confused with the Tsallis q-exponential.

Definition

The q-exponential ${\displaystyle e_{q}(z)}$ is defined as

${\displaystyle e_{q}(z)=\sum _{n=0}^{\infty }{\frac {z^{n}}{[n]_{q}!}}=\sum _{n=0}^{\infty }{\frac {z^{n}(1-q)^{n}}{(q;q)_{n}}}=\sum _{n=0}^{\infty }z^{n}{\frac {(1-q)^{n}}{(1-q^{n})(1-q^{n-1})\cdots (1-q)}}}$

where ${\displaystyle [n]_{q}!}$ is the q-factorial and

${\displaystyle (q;q)_{n}=(1-q^{n})(1-q^{n-1})\cdots (1-q)}$

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

${\displaystyle \left({\frac {d}{dz}}\right)_{q}e_{q}(z)=e_{q}(z)}$

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

${\displaystyle \left({\frac {d}{dz}}\right)_{q}z^{n}=z^{n-1}{\frac {1-q^{n}}{1-q}}=[n]_{q}z^{n-1}.}$

Here, ${\displaystyle [n]_{q}}$ is the q-bracket. For other definitions of the q-exponential function, see Exton (1983) harvtxt error: no target: CITEREFExton1983 (help), Ismail & Zhang (1994) harvtxt error: no target: CITEREFIsmailZhang1994 (help), Suslov (2003) harvtxt error: no target: CITEREFSuslov2003 (help) and Cieslinski (2011) harvtxt error: no target: CITEREFCieslinski2011 (help).

Properties

For real ${\displaystyle q>1}$, the function ${\displaystyle e_{q}(z)}$ is an entire function of ${\displaystyle z}$. For ${\displaystyle q<1}$, ${\displaystyle e_{q}(z)}$ is regular in the disk ${\displaystyle |z|<1/(1-q)}$.

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

Addition Formula

If ${\displaystyle xy=qyx}$, ${\displaystyle e_{q}(x)e_{q}(y)=e_{q}(x+y)}$ holds.

Relations

For ${\displaystyle -1<q<1}$, a function that is closely related is ${\displaystyle E_{q}(z).}$ It is a special case of the basic hypergeometric series,

${\displaystyle E_{q}(z)=\;_{1}\phi _{1}\left({\scriptstyle {0 \atop 0}}\,;\,z\right)=\sum _{n=0}^{\infty }{\frac {q^{\binom {n}{2}}(-z)^{n}}{(q;q)_{n}}}=\prod _{n=0}^{\infty }(1-q^{n}z)=(z;q)_{\infty }.}$

Clearly,

${\displaystyle \lim _{q\to 1}E_{q}\left(z(1-q)\right)=\lim _{q\to 1}\sum _{n=0}^{\infty }{\frac {q^{\binom {n}{2}}(1-q)^{n}}{(q;q)_{n}}}(-z)^{n}=e^{-z}.~}$

Relation with Dilogarithm

${\displaystyle e_{q}(x)}$ has the following infinite product representation:

${\displaystyle e_{q}(x)=\left(\prod _{k=0}^{\infty }(1-q^{k}(1-q)x)\right)^{-1}.}$

On the other hand, ${\displaystyle \log(1-x)=-\sum _{n=1}^{\infty }{\frac {x^{n}}{n}}}$ holds. When ${\displaystyle |q|<1}$,

${\displaystyle \log e_{q}(x)=-\sum _{k=0}^{\infty }\log(1-q^{k}(1-q)x)=\sum _{k=0}^{\infty }\sum _{n=1}^{\infty }{\frac {(q^{k}(1-q)x)^{n}}{n}}=\sum _{n=1}^{\infty }{\frac {((1-q)x)^{n}}{(1-q^{n})n}}={\frac {1}{1-q}}\sum _{n=1}^{\infty }{\frac {((1-q)x)^{n}}{[n]_{q}n}}.}$

By taking the limit ${\displaystyle q\to 1}$,

${\displaystyle \lim _{q\to 1}(1-q)\log e_{q}(x/(1-q))=\mathrm {Li} _{2}(x),}$

where ${\displaystyle \mathrm {Li} _{2}(x)}$ is the dilogarithm.

References

• Exton, H. (1983), q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis Horwood, ISBN 0853124914, ISBN 0470274530, ISBN 978-0470274538
• Gasper, G. & Rahman, M. (2004), Basic Hypergeometric Series, Cambridge University Press, ISBN 0521833574
• Ismail, M. E. H. (2005), Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press.
• Ismail, M. E. H. & Zhang, R. (1994), “Diagonalization of certain integral operators,” Advances in Math. 108, 1–33.
• Ismail, M.E.H. Rahman, M. & Zhang, R. (1996), Diagonalization of certain integral operators II, J. Comp. Appl. Math. 68, 163-196.
• Jackson, F. H. (1908), "On q-functions and a certain difference operator", Transactions of the Royal Society of Edinburgh, 46, 253-281.