{"history":[],"article":{"id":"b19edc87-f65d-4014-bfa0-03e91dc216cd","title":"Q-exponential","slug":"q-exponential","content":":::summary\n:::\n\n# \nIn [combinatorial](https://bliptext.com/articles/combinatorics) [mathematics](https://bliptext.com/articles/mathematics), a q-exponential is a q-analog of the [exponential function](https://bliptext.com/articles/exponential-function),\nnamely the [eigenfunction](https://bliptext.com/articles/eigenfunction) of a q-derivative.\nThere are many q-derivatives, for example, the classical q-derivative, the [Askey–Wilson operator](https://bliptext.com/articles/askey-wilson-polynomials), etc. Therefore, unlike the classical exponentials, q-exponentials are not unique.\nFor example, e_q(z) is the q-exponential corresponding to the classical q-derivative while are eigenfunctions of the Askey–Wilson operators.\nThe q-exponential is also known as the [quantum dilogarithm](https://bliptext.com/articles/quantum-dilogarithm).\n\n# Definition\nThe q-exponential e_q(z) is defined as\nwhere [n]!_q is the q-factorial and\nis the q-Pochhammer symbol.\nThat this is the q-analog of the exponential follows from the property\nwhere the derivative on the left is the q-derivative.\nThe above is easily verified by considering the q-derivative of the [monomial](https://bliptext.com/articles/monomial)\nHere, [n]_q is the q-bracket.\nFor other definitions of the q-exponential function, see, , and.\n\n# Properties\nFor real q>1, the function e_q(z) is an [entire function](https://bliptext.com/articles/entire-function) of z. For q<1, e_q(z) is regular in the disk |z|<1/(1-q).\nNote the inverse,.\n\n## Addition Formula\nThe analogue of does not hold for real numbers x and y. However, if these are operators satisfying the commutation relation xy=qyx, then holds true.\n\n# Relations\nFor -11, the function e_q(z) is an [entire function](https://bliptext.com/articles/entire-function) of z. For q<1, e_q(z) is regular in the disk |z|<1/(1-q).\nNote the inverse,.\n\n## Addition Formula\nThe analogue of does not hold for real numbers x and y. However, if these are operators satisfying the commutation relation xy=qyx, then holds true.\n\n# Relations\nFor -1

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