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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 Exton (1983), Ismail & Zhang (1994) and Suslov (2003).


For real , the function is an entire function of . For , is regular in the disk .

Note the inverse, .


For , a function that is closely related is It is a special case of the basic hypergeometric series,



  • 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.
  • Jackson, F. H. (1908), "On q-functions and a certain difference operator", Transactions of the Royal Society of Edinburgh, 46, 253-281.