Digital Library of Mathematical Functions
The Gamma Function -- Barnes G-Function (Double Gamma Function)

### Barnes G -Function (Double Gamma Function)

G ( z + 1 ) = Γ ( z ) G ( z ) , G ( 1 ) = 1 ,
G ( n ) = ( n - 2 ) ! ( n - 3 ) ! ⋯ 1 !
n = 2 , 3 , …
G ( z + 1 ) = ( 2 π ) z 2 exp ( - 1 2 z ( z + 1 ) - 1 2 γ z 2 ) × ∏ k = 1 ∞ ( ( 1 + z k ) k exp ( - z + z 2 2 k ) )
Ln G ( z + 1 ) = 1 2 z ln ( 2 π ) - 1 2 z ( z + 1 ) + z Ln Γ ( z + 1 ) - ∫ 0 z Ln Γ ( t + 1 ) ⅆ t

The Ln 's have their principal values on the positive real axis and are continued via continuity.

When z → ∞ in | ph z | ≤ π - δ ( < π )

Ln G ( z + 1 ) ∼ 1 4 z 2 + z Γ ( z + 1 ) - ( 1 2 z ( z + 1 ) + 1 12 ) Ln z - ln A + ∑ k = 1 ∞ B 2 k + 2 2 k ( 2 k + 1 ) ( 2 k + 2 ) z 2 k

see Ferreira and López(2001) . This reference also provides bounds for the error term. Here B 2 k + 2 is the Bernoulli number, and A is Glaisher's constant, given by

A = ⅇ C = 1.28242 71291 00622 63687 …

where

C = lim n → ∞ ( ∑ k = 1 n k ln k - ( 1 2 n 2 + 1 2 n + 1 12 ) ln n + 1 4 n 2 ) = γ + ln ( 2 π ) 12 - ζ ′ ( 2 ) 2 π 2 = 1 12 - ζ ′ ( - 1 )

and ζ ′ is the derivative of the zeta function

For Glaisher's constant see also Greene and Knuth(1982) (p. 100).