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, is the q-exponential corresponding to the classical q-derivative while are eigenfunctions of the Askey-Wilson operators. The q-exponential is defined as where is the q-factorial and is the q-Pochhammer symbol. That this is the q-analog of the exponential follows from the property where the derivative on the left is the q-derivative. The above is easily verified by considering the q-derivative of the monomial Here, is the q-bracket. For other definitions of the q-exponential function, see , , and . For real , the function is an entire function of . For , is regular in the disk . Note the inverse, . The analogue of does not hold for real numbers and . However, if these are operators satisfying the commutation relation , then holds true. For , a function that is closely related is It is a special case of the basic hypergeometric series, Clearly, has the following infinite product representation: On the other hand, holds. When , By taking the limit , where is the dilogarithm. Quantum dilogarithm The Q-exponential function is also known as the quantum dilogarithm.