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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)
=2int_0^1(t^(p-1)dt)/((1+t)^(2p))
(4)
=2/pproduct_(n=1)^(infty)(n(n+2p))/((n+p)(n+p)).
(5)

The integral representation requires Re(p)>0, where Re(p) is the real part of p. The other identities are interpreted by analytic continuation as identities of meromorphic functions, with limits at removable singularities. With p=1/2, the infinite product gives the Wallis formula. For p=1, 2, ... the first few values are 1, 1/6, 1/30, 1/140, 1/630, 1/2772, ..., whose denominators are (2p-1)!/(p-1)!^2 (OEIS A002457).

For positive real numbers a and b with p=a/b,

beta(a/b)=b2^(1-2a/b)J(a,b),
(6)

where

J(a,b)=int_0^1(t^(a-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)beta(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. "Fast Evaluation of the Gamma Function for Small Rational Fractions Using Complete Elliptic Integrals of the First Kind." IMA J. Numer. Anal. 12, 519-526, 1992. https://doi.org/10.1093/imanum/12.4.519.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 Resource. https://mathworld.wolfram.com/CentralBetaFunction.html

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