Supriya Ghosh (Editor)

Q gamma function

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In q-analog theory, the q-gamma function, or basic gamma function, is a generalization of the ordinary Gamma function closely related to the double gamma function. It was introduced by Jackson (1905). It is given by

Γ q ( x ) = ( 1 − q ) 1 − x ∏ n = 0 ∞ 1 − q n + 1 1 − q n + x = ( 1 − q ) 1 − x ( q ; q ) ∞ ( q x ; q ) ∞

when |q|<1, and

Γ q ( x ) = ( q − 1 ; q − 1 ) ∞ ( q − x ; q − 1 ) ∞ ( q − 1 ) 1 − x q ( x 2 )

if |q|>1. Here (·;·)∞ is the infinite q-Pochhammer symbol. It satisfies the functional equation

Γ q ( x + 1 ) = 1 − q x 1 − q Γ q ( x ) = [ x ] q Γ q ( x )

For non-negative integers n,

Γ q ( n ) = [ n − 1 ] q !

where [·]q! is the q-factorial function. Alternatively, this can be taken as an extension of the q-factorial function to the real number system.

The relation to the ordinary gamma function is made explicit in the limit

lim q → 1 ± Γ q ( x ) = Γ ( x ) .

Due to I. Mező, the q-analogue of the Raabe formula exists, at least if we use the q-gamma function when |q|>1. With this restriction

∫ 0 1 log ⁡ Γ q ( x ) d x = ζ ( 2 ) log ⁡ q + log ⁡ q − 1 q 6 + log ⁡ ( q − 1 ; q − 1 ) ∞ ( q > 1 ) .

El Bachraoui considered the case 0<q<1 and proved that

∫ 0 1 log ⁡ Γ q ( x ) d x = 1 2 log ⁡ ( 1 − q ) − ζ ( 2 ) log ⁡ q + log ⁡ ( q ; q ) ∞ ( 0 < q < 1 ) .

References

Q-gamma function Wikipedia
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