Q-exponential Last updated June 10, 2025 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 . [ 1] [ 2]
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) .
References Cieśliński, Jan L. (2011). "Improved q-exponential and q-trigonometric functions" . Applied Mathematics Letters . 24 (12): 2110– 2114. arXiv : 1006.5652 . doi : 10.1016/j.aml.2011.06.009 . S2CID 205496812 . Exton, Harold (1983). q-Hypergeometric Functions and Applications . New York: Halstead Press, Chichester: Ellis Horwood. ISBN 0853124914 . Gasper, George ; Rahman, Mizan Rahman (2004). Basic Hypergeometric Series . Cambridge University Press. ISBN 0521833574 . Ismail, Mourad E. H. (2005). Classical and Quantum Orthogonal Polynomials in One Variable . Cambridge University Press. doi :10.1017/CBO9781107325982 . ISBN 9780521782012 . Ismail, Mourad E. H. ; Zhang, Ruiming (1994). "Diagonalization of certain integral operators" . Advances in Mathematics . 108 (1): 1– 33. doi : 10.1006/aima.1994.1077 . Ismail, Mourad E. H. ; Rahman, Mizan ; Zhang, Ruiming (1996). "Diagonalization of certain integral operators II" . Journal of Computational and Applied Mathematics . 68 (1– 2): 163– 196. CiteSeerX 10.1.1.234.4251 . doi : 10.1016/0377-0427(95)00263-4 . Jackson, F. H. (1909). "On q-functions and a certain difference operator". Transactions of the Royal Society of Edinburgh . 46 (2): 253– 281. doi :10.1017/S0080456800002751 . S2CID 123927312 . This page is based on this
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