Central Beta Function

Min   Max       Re       Im

The central beta function is defined by

(1)

where  is the beta function. It satisfies the identities

			(2)
			(3)
			(4)
			(5)

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."