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Central Beta Function


\begin{figure}\begin{center}\BoxedEPSF{CentralBetaFunctionReIm.epsf scaled 810}\end{center}\end{figure}

The central beta function is defined by

\beta(p)\equiv B(p,p),
\end{displaymath} (1)

where $B(p,q)$ is the Beta Function. It satisfies the identities
$\displaystyle \beta(p)$ $\textstyle =$ $\displaystyle 2^{1-2p}B(p, {\textstyle{1\over 2}})$ (2)
  $\textstyle =$ $\displaystyle 2^{1-2p}\cos(\pi p)B({\textstyle{1\over 2}}-p, p)$ (3)
  $\textstyle =$ $\displaystyle \int_0^1 {t^p\,dt\over (1+t)^{2p}}$ (4)
  $\textstyle =$ $\displaystyle {2\over p}\prod_{n=1}^\infty {n(n+2p)\over (n+p)(n+p)}.$ (5)

With $p=1/2$, the latter gives the Wallis Formula. When $p=a/b$,
\end{displaymath} (6)

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

The central beta function satisfies
\end{displaymath} (8)

(1-2x)\beta(1-x)\beta(x)=2\pi\cot(\pi x)
\end{displaymath} (9)

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

\beta(x)\beta(x+{\textstyle{1\over 2}})=2^{4x+1}\pi\beta(2x)\beta(2x+{\textstyle{1\over 2}}).
\end{displaymath} (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).
\end{displaymath} (12)

See also Beta Function, Regularized Beta Function


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.

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© 1996-9 Eric W. Weisstein