Difference Equation, Functional Equation and Special Values
The Barnes G-function satisfies the difference equation
with normalisation G(1) = 1. The difference equation implies that G takes the following values at integer arguments:
and thus
where Γ denotes the Gamma function and K denotes the K-function. The difference equation uniquely defines the G function if the convexity condition: is added.
The difference equation for the G function and the functional equation for the Gamma function yield the following functional equation for the G function, originally proved by Hermann Kinkelin:
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