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 ) {\displaystyle e_{q}(z)} is the q-exponential corresponding to the classical q-derivative while E q ( z ) {\displaystyle {\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 ) {\displaystyle e_{q}(z)} is defined as
e q ( z ) = ∑ n = 0 ∞ z n [ n ] ! q = ∑ n = 0 ∞ z n ( 1 − q ) n ( q ; q ) n = ∑ n = 0 ∞ z n ( 1 − q ) n ( 1 − q n ) ( 1 − q n − 1 ) ⋯ ( 1 − q ) {\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 [ n ] ! q {\displaystyle [n]!_{q}} is the q-factorial and
( q ; q ) n = ( 1 − q n ) ( 1 − q n − 1 ) ⋯ ( 1 − q ) {\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
( d d z ) q e q ( z ) = e q ( z ) {\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
( d d z ) q z n = z n − 1 1 − q n 1 − q = [ n ] q z n − 1 . {\displaystyle \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 {\displaystyle [n]_{q}} is the q-bracket. For other definitions of the q-exponential function, see Exton (1983), Ismail & Zhang (1994), and Cieśliński (2011).
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