Digital Library of Mathematical Functions
The Gamma Function -- q-Gamma and Beta Functions

q -Gamma and Beta Functions

q -Factorials

( a ; q ) n = ∏ k = 0 n - 1 ( 1 - a q k )
n = 0 , 1 , 2 , …
n ! q = 1 ( 1 + q ) ⋯ ( 1 + q + … + q n - 1 ) = ( q ; q ) n ( 1 - q ) - n

When | q | < 1 ,

( a ; q ) ∞ = ∏ k = 0 ∞ ( 1 - a q k )

q-Gamma Function

When 0 < q < 1 ,

Γ q ( z ) = ( q ; q ) ∞ ( 1 - q ) 1 - z ( q z ; q ) ∞
Γ q ( 1 ) = Γ q ( 2 ) = 1
n ! q = Γ q ( n + 1 )
Γ q ( z + 1 ) = 1 - q z 1 - q Γ q ( z )

Also, ln Γ q ( x ) is convex for x > 0 , and the analog of the Bohr-Mollerup theorem holds.

If 0 < q < r < 1 , then

Γ q ( x ) < Γ r ( x )

when 0 < x < 1 or when x > 2 , and

Γ q ( x ) > Γ r ( x )

when 1 < x < 2 .

lim q → 1 - Γ q ( z ) = Γ ( z )

For generalized asymptotic expansions of ln Γ q ( z ) as | z | → ∞ see Olde Daalhuis(1994) and Moak(1984) .

q -Beta Function

B q ( a , b ) = Γ q ( a ) Γ q ( b ) Γ q ( a + b )
B q ( a , b ) = ∫ 0 1 t a - 1 ( t q ; q ) ∞ ( t q b ; q ) ∞ ⅆ q t
0 < q < 1 , ℜ a > 0 , ℜ b > 0 .