“Q-exponential”的意思、由来-开放百科全书

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Q-exponential

释义

Definition
Properties
Addition Formula
Relations
Relation with Dilogarithm
References

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.

Definition

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 {{harvtxt|Exton|1983}}, {{harvtxt|Ismail|Zhang|1994}}, {{harvtxt|Suslov|2003}} and {{harvtxt|Cieslinski|2011}}.

Properties

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

Note the inverse, .

Addition Formula

If , holds.

Relations

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

Clearly,

Relation with Dilogarithm

has the following infinite product representation:

On the other hand, holds.

When ,

By taking the limit ,

where is the dilogarithm.

References

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.

2 : Q-analogs|Exponentials

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