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nLab > Latest Changes: Barnes G-function

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeJun 1st 2021
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Author: Urs
Format: MarkdownItexPage created, but author did not leave any comments. <a href="https://ncatlab.org/nlab/revision/Barnes+G-function/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Barnes+G-function">current</a>

Page created, but author did not leave any comments.

v1, current
- CommentRowNumber2.
- CommentAuthorDavid_Corfield
- CommentTimeJun 1st 2021
- (edited Jun 1st 2021)
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Author: David_Corfield
Format: MarkdownItexLet's see, we should be able to show this. If $N = 2M$, $$ \underoverset {j = 1} {N} {\prod} \Gamma(j/2) \;=\; \underoverset {j = 1} {M} {\prod} \Gamma(j) \cdot \underoverset {j = 1} {M} {\prod} \Gamma(j - \frac{1}{2}) \;=\; G(M+1) \cdot G(M +1/2) /G(1/2) \;=\; G(N/2 + 1) \cdot G(N/2 + 1/2)/G(1/2). $$ If $N = 2M+1$, $$ \underoverset {j = 1} {N} {\prod} \Gamma(j/2) \;=\; \underoverset {j = 1} {M} {\prod} \Gamma(j) \cdot \underoverset {j = 1} {M+1} {\prod} \Gamma(j - \frac{1}{2}) \;=\; G(M+1) \cdot G(M +3/2) /G(1/2) \;=\; G(N/2 + 1/2) \cdot G(N/2 + 1)/G(1/2). $$

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

If $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>=</mo><mn>2</mn><mi>M</mi></mrow><annotation encoding="application/x-tex">N = 2M</annotation></semantics></math>$ ,
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munderover><mo lspace="0.16667em" rspace="0.16667em">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mi>Γ</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">/</mo><mn>2</mn><mo stretchy="false">)</mo><mspace width="0.27778em"/><mo>=</mo><mspace width="0.27778em"/><munderover><mo lspace="0.16667em" rspace="0.16667em">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>M</mi></munderover><mi>Γ</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo>⋅</mo><munderover><mo lspace="0.16667em" rspace="0.16667em">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>M</mi></munderover><mi>Γ</mi><mo stretchy="false">(</mo><mi>j</mi><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">)</mo><mspace width="0.27778em"/><mo>=</mo><mspace width="0.27778em"/><mi>G</mi><mo stretchy="false">(</mo><mi>M</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>⋅</mo><mi>G</mi><mo stretchy="false">(</mo><mi>M</mi><mo>+</mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>G</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn><mo stretchy="false">)</mo><mspace width="0.27778em"/><mo>=</mo><mspace width="0.27778em"/><mi>G</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">/</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>⋅</mo><mi>G</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">/</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>G</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> \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). </annotation></semantics></math>$
If $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>=</mo><mn>2</mn><mi>M</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N = 2M+1</annotation></semantics></math>$ ,
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munderover><mo lspace="0.16667em" rspace="0.16667em">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mi>Γ</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">/</mo><mn>2</mn><mo stretchy="false">)</mo><mspace width="0.27778em"/><mo>=</mo><mspace width="0.27778em"/><munderover><mo lspace="0.16667em" rspace="0.16667em">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>M</mi></munderover><mi>Γ</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo>⋅</mo><munderover><mo lspace="0.16667em" rspace="0.16667em">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>M</mi><mo>+</mo><mn>1</mn></mrow></munderover><mi>Γ</mi><mo stretchy="false">(</mo><mi>j</mi><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">)</mo><mspace width="0.27778em"/><mo>=</mo><mspace width="0.27778em"/><mi>G</mi><mo stretchy="false">(</mo><mi>M</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>⋅</mo><mi>G</mi><mo stretchy="false">(</mo><mi>M</mi><mo>+</mo><mn>3</mn><mo stretchy="false">/</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>G</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn><mo stretchy="false">)</mo><mspace width="0.27778em"/><mo>=</mo><mspace width="0.27778em"/><mi>G</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">/</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn><mo stretchy="false">)</mo><mo>⋅</mo><mi>G</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">/</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>G</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> \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). </annotation></semantics></math>$
- CommentRowNumber3.
- CommentAuthorDavid_Corfield
- CommentTimeJun 1st 2021
- PermaLink
Author: David_Corfield
Format: MarkdownItexAdded that calculation. <a href="https://ncatlab.org/nlab/revision/diff/Barnes+G-function/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/Barnes+G-function/2">v2</a>, <a href="https://ncatlab.org/nlab/show/Barnes+G-function">current</a>

Added that calculation.

diff, v2, current
- CommentRowNumber4.
- CommentAuthorUrs
- CommentTimeJun 1st 2021
- PermaLink
Author: Urs
Format: MarkdownItexNice. I added some formatting and mentioning of the initial value $G(1) = 1$. <a href="https://ncatlab.org/nlab/revision/diff/Barnes+G-function/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/Barnes+G-function/3">v3</a>, <a href="https://ncatlab.org/nlab/show/Barnes+G-function">current</a>

Nice. I added some formatting and mentioning of the initial value $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">G(1) = 1</annotation></semantics></math>$ .

diff, v3, current

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nLab > Latest Changes: Barnes G-function