A generalization of Gauss' theorem on arithmetic-geometric means
The author establishes the following theorem. Let \(S^{n-1}\) denote the unit sphere in \({\mathbb{R}}^ n\), and let \(d\omega_{n-1}\) be the volume element of \(S^{n-1}\) such that the volume of \(S^{n-1}\) is unity. Moreover, let \(0<a\leq b\) and \(a_ 1=\sqrt{(ab)}\), \(b_ 1=\frac{a+b}{2}\). Then, \[ \int_{S^{2p-1}}\frac{d\omega_{2p- 1}}{[a^ 2(x_ 1^ 2+...+x^ 2_ p)+b^ 2(x^ 2_{p+1}+...+x^ 2_{2p})]^ s}= \] \[ \frac{2\Gamma(frac{1}{2}p+frac{1}{2})}{\Gamma(p- s)\Gamma(s-frac{1}{2}p+frac{1}{2})}\int^{\pi /2}_{0}\frac{(\sin \theta)^{2p-2s-1}(\cos \theta)^{2s-p}d\theta}{(a^ 2_ 1\cos^ 2\theta +b^ 2_ 1\sin^ 2\theta)^ s}, \] where \(p>Re s>frac{1}{2}(p-1).\) The author utilizes some classical properties of Gegenbauer polynomials, a quadratic transformation of \({}_ 2F_ 1\), and some results which he has given recently [Nagoya Math. J. 92, 133-144 (1983; reviewed above)]. For \(p=1\), \(s=frac{1}{2}\), the theorem reduces to a result due to Gauß.
scientific article; zbMATH DE number 3844136