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 also known as the quantum dilogarithm.[1][2]

Definition

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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 Cieśliński (2011).

Properties

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

Note the inverse,  .

Addition Formula

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The analogue of   does not hold for real numbers   and  . However, if these are operators satisfying the commutation relation  , then   holds true.[3]

Relations

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

 

Clearly,

 

Relation with Dilogarithm

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  has the following infinite product representation:

 

On the other hand,   holds. When  ,

 

By taking the limit  ,

 

where   is the dilogarithm.

References

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  1. ^ Zudilin, Wadim (14 March 2006). "Quantum dilogarithm" (PDF). wain.mi.ras.ru. Retrieved 16 July 2021.
  2. ^ Faddeev, L.d.; Kashaev, R.m. (1994-02-20). "Quantum dilogarithm". Modern Physics Letters A. 09 (5): 427–434. arXiv:hep-th/9310070. Bibcode:1994MPLA....9..427F. doi:10.1142/S0217732394000447. ISSN 0217-7323. S2CID 119124642.
  3. ^ Kac, V.; Cheung, P. (2011). Quantum Calculus. Springer. p. 31. ISBN 978-1461300724.