The central beta function is defined by
(1)
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where is the beta function. It satisfies the identities
(2)
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(3)
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(4)
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(5)
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With , the latter gives the Wallis formula. For , 2, ... the first few values are 1, 1/6, 1/30, 1/140, 1/630, 1/2772, ... (OEIS A002457), which have denominators .
When ,
(6)
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where
(7)
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The central beta function satisfies
(8)
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(9)
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(10)
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(11)
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For an odd positive integer, the central beta function satisfies the identity
(12)
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