Central Beta Function

The central beta function is defined by

$$\beta(p)\equiv B(p,p),$$ (1)

where $B(p,q)$ is the Beta Function. It satisfies the identities

$$\beta(p) = 2^{1-2p}B(p, {\textstyle{1\over 2}})$$ (2)

$$= 2^{1-2p}\cos(\pi p)B({\textstyle{1\over 2}}-p, p)$$ (3)

$$= \int_0^1 {t^p\,dt\over (1+t)^{2p}}$$ (4)

$$= {2\over p}\prod_{n=1}^\infty {n(n+2p)\over (n+p)(n+p)}.$$ (5)

With $p=1/2$, the latter gives the Wallis Formula. When $p=a/b$,

$$b\beta(a/b)=2^{1-2a/b}J(a,b),$$ (6)

where

$$J(a,b)\equiv \int_0^1 {t^{\alpha-1}\,dt\over\sqrt{1-t^b}}.$$ (7)

The central beta function satisfies

$$(2+4x)\beta(1+x)=x\beta(x)$$ (8)

$$(1-2x)\beta(1-x)\beta(x)=2\pi\cot(\pi x)$$ (9)

$$\beta({\textstyle{1\over 2}}-x)=2^{4x-1}\tan(\pi x)\beta(x)$$ (10)

$$\beta(x)\beta(x+{\textstyle{1\over 2}})=2^{4x+1}\pi\beta(2x)\beta(2x+{\textstyle{1\over 2}}).$$ (11)

For $p$ an Odd Positive Integer, the central beta function satisfies the identity

$$\beta(px)={1\over\sqrt{p}}\prod_{k=1}^{(p-1)/2}{2x+{2k-1\over p}\over 2\pi}\prod_{k=0}^{p-1}\beta\left({x+{k\over p}}\right).$$ (12)

See also Beta Function, Regularized Beta Function

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.