Central Beta Function
المؤلف:
Borwein, J. M. and Zucker, I. J.
المصدر:
"Elliptic Integral Evaluation of the Gamma Function at Rational Values of Small Denominators." IMA J. Numerical Analysis 12
الجزء والصفحة:
...
21-5-2019
2211
Central Beta Function
The central beta function is defined by
 |
(1)
|
where 
is the beta function. It satisfies the identities
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)
|
where
 |
(7)
|
The central beta function satisfies
 |
(8)
|
 |
(9)
|
 |
(10)
|
 |
(11)
|
For 
an odd positive integer, the central beta function satisfies the identity
 |
(12)
|
REFERENCES:
Borwein, J. M. and Zucker, I. J. "Elliptic Integral Evaluation of the Gamma Function at Rational Values of Small Denominators." IMA J. Numerical Analysis 12, 519-526, 1992.
Sloane, N. J. A. Sequence A002457/M4198 in "The On-Line Encyclopedia of Integer Sequences."
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