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nLab > Latest Changes: Barnes G-function

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 1st 2021

    Page created, but author did not leave any comments.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 1st 2021
    • (edited Jun 1st 2021)

    Let’s see, we should be able to show this.

    If N=2MN = 2M,

    ∏j=1NΓ(j/2)=∏j=1MΓ(j)⋅∏j=1MΓ(j−12)=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).

    If N=2M+1N = 2M+1,

    ∏j=1NΓ(j/2)=∏j=1MΓ(j)⋅∏j=1M+1Γ(j−12)=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).
    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 1st 2021

    Added that calculation.

    diff, v2, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 1st 2021

    Nice. I added some formatting and mentioning of the initial value G(1)=1G(1) = 1.

    diff, v3, current

1 to 4 of 4

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