Q exponential - Alchetron, The Free Social Encyclopedia

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 ) is the q-exponential corresponding to the classical q-derivative while E q ( z ) are eigenfunctions of the Askey-Wilson operators.

Definition

The q-exponential 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 )

where [ n ] q ! is the q-factorial and

( q ; q ) n = ( 1 − q n ) ( 1 − q n − 1 ) ⋯ ( 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 )

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 .

Here, [ n ] q is the q-bracket.

Properties

For real q > 1 , the function e q ( z ) is an entire function of z. For q < 1 , e q ( z ) is regular in the disk | z | < 1 / ( 1 − q ) .

Note the inverse, e q ( z ) e 1 / q ( − z ) = 1 .

Relations

For − 1 < q < 1 , a function that is closely related is E q ( z ) . It is a special case of the basic hypergeometric series,

E q ( z ) = 0 ϕ 1 ( − ; 0 , − z ) = ∑ n = 0 ∞ q ( n 2 ) z n ( q ; q ) n = ∏ n = 0 ∞ ( 1 − q n z ) .

Clearly,

lim q → 1 E q ( z ( 1 − q ) ) = lim q → 1 ∑ n = 0 ∞ q ( n 2 ) ( 1 − q ) n ( q ; q ) n z n = e z .

References

Q-exponential Wikipedia

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Contents

Definition

Properties

Relations

References