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Central Beta Function

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The central beta function is defined by

beta(p)=B(p,p),
(1)

where B(p,q) is the beta function. It satisfies the identities

beta(p)=2^(1-2p)B(p,1/2)
(2)
=2^(1-2p)cos(pip)B(1/2-p,p)
(3)
=int_0^1(t^pdt)/((1+t)^(2p))
(4)
=2/pproduct_(n=1)^(infty)(n(n+2p))/((n+p)(n+p)).
(5)

With p=1/2, the latter gives the Wallis formula. For p=1, 2, ... the first few values are 1, 1/6, 1/30, 1/140, 1/630, 1/2772, ... (OEIS A002457), which have denominators (n-1)!^2/(2n-1)!.

When p=a/b,

bbeta(a/b)=2^(1-2a/b)J(a,b),
(6)

where

J(a,b)=int_0^1(t^(alpha-1)dt)/(sqrt(1-t^b)).
(7)

The central beta function satisfies

(2+4x)beta(1+x)=xbeta(x)
(8)
(1-2x)beta(1-x)beta(x)=2picot(pix)
(9)
beta(1/2-x)=2^(4x-1)tan(pix)beta(x)
(10)
beta(x)beta(x+1/2)=2^(4x+1)pibeta(2x)beta(2x+1/2).
(11)

For p an odd positive integer, the central beta function satisfies the identity

beta(px)=1/(sqrt(p))product_(k=1)^((p-1)/2)(2x+(2k-1)/p)/(2pi)product_(k=0)^(p-1)beta(x+k/p).
(12)

See also

Beta Function, Regularized Beta Function

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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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Central Beta Function

Cite this as:

Weisstein, Eric W. "Central Beta Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CentralBetaFunction.html

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