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Barnes G-function

Contents

Contents

Idea

Much like the Gamma function generalizes the functional equation

Γ(z+1)=zΓ(z) \Gamma(z + 1) \;=\; z \, \Gamma(z)

to non-integer values of zz, so the Barnes GG-function G()G(-) corresponds to the functional equation

(1)G(z+1)=Γ(z)G(z). G(z + 1) \;=\; \Gamma(z) \, G(z) \,.

Just as for z=nz =n \in \mathbb{N}, Γ(n+1)=n!\Gamma(n+1) = n!, so G(n+2)=n!(n1)!1!G(n+2) = n!(n-1)! \cdots 1!.

Definition

(…)

Properties

Special values

(2)G(1)=1 G(1) \;=\; 1
G(1/2)=2 1/24e 32ζ (1)π 1/4. G(1/2) \;=\; 2^{1/24} \cdot e^{ \tfrac{3}{2} \zeta^'(-1) } \cdot \pi^{ - 1/4 } \,.

(WP here)

Stirling-like asymptotic expansion

lnG(1+z)=z 2(12ln(z)34)+12ln(2π)z112ln(z)+ζ (1)+𝒪(z 1), ln G(1 + z) \;=\; z^2 \left( \tfrac{1}{2} ln(z) - \tfrac{3}{4} \right) + \tfrac{1}{2} ln(2 \pi) z - \tfrac{1}{12} ln(z) + \zeta^'(-1) + \mathcal{O}(z^{-1}) \,,

where ζ\zeta denotes the Riemann zeta function.

(e.g. WP here, WMW (14))

Relations to the Gamma-function

A version of the Gauss multiplication formula for the Gamma function:

Proposition

j=1NΓ(j/2)=G(N/2+1)G(N/2+1/2)G(1/2). \underoverset {j = 1} {N} {\prod} \Gamma(j/2) \;=\; \frac { G(N/2+ 1) \cdot G(N/2 + 1/2) } { G(1/2) } \,.

(Kotěšovec 13, p. 2)
Proof

By repeated use of the translation formula (1) and using the initial value G(1)=1G(1) = 1 (2):

In the case that N=2MN = 2M in an even number:

j=1NΓ(j/2)=j=1MΓ(j)j=1MΓ(j12)=G(M+1)G(M+1/2)/G(1/2)=G(N/2+1)G(N/2+1/2)/G(1/2). \underoverset {j = 1} {N} {\prod} \Gamma(j/2) \;=\; \underoverset {j = 1} {M} {\prod} \Gamma(j) \cdot \underoverset {j = 1} {M} {\prod} \Gamma(j - \frac{1}{2}) \;=\; G(M+1) \cdot G(M +1/2) /G(1/2) \;=\; G(N/2 + 1) \cdot G(N/2 + 1/2)/G(1/2).

In the case that N=2M+1N = 2M+1 is an odd number:

j=1NΓ(j/2)=j=1MΓ(j)j=1M+1Γ(j12)=G(M+1)G(M+3/2)/G(1/2)=G(N/2+1/2)G(N/2+1)/G(1/2). \underoverset {j = 1} {N} {\prod} \Gamma(j/2) \;=\; \underoverset {j = 1} {M} {\prod} \Gamma(j) \cdot \underoverset {j = 1} {M+1} {\prod} \Gamma(j - \frac{1}{2}) \;=\; G(M+1) \cdot G(M +3/2) /G(1/2) \;=\; G(N/2 + 1/2) \cdot G(N/2 + 1)/G(1/2).

References

See also:

In the context of counting of standard Young tableaux of bounded height:

  • Václav Kotěšovec, Asymptotic of Young tableaux of bounded height, 2013 (pdf, pdf)

Last revised on June 1, 2021 at 09:34:19. See the history of this page for a list of all contributions to it.