Barnard, Roger W.; Pearce, Kent; Richards, Kendall C. A monotonicity property involving \(_3F_2\) and comparisons of the classical approximations of elliptical arc length. (English) Zbl 0983.33006 SIAM J. Math. Anal. 32, No. 2, 403-419 (2000). Authors’ abstract: Conditions are determined under which \(_3 F_2(-n, a, b; a+b+2,\varepsilon -n+1;1)\) is a monotone function of \(n\) satisfying \( _3 F_2(-n, a, b; a+b+2,\varepsilon -n+1;1)\geq ab _2 F_1(a, b; a+b+2;1) \). Motivated by a conjecture of M. Vuorinen [K. Srinivasa Rao (ed.) et al., Special functions and differential equations, Proceedings of a workshop, WSSF ‘97, Madras, India, January 13–24, 1997. New Delhi: Allied Publishers Private Ltd., 119–126 (1998; Zbl 0948.30024)], the corollary that \(_3 F_2(-n, -1/2, -1/2; 1,\varepsilon -n+1;1)\geq 4/\pi\), for \(1>\varepsilon >1/4\) and \(n \geq 2\), is used to determine surprising hierarchical relationships among the 13 known historical approximations of the arc length of an ellipse. This complete list of inequalities compares the Maclaurin series coefficients of \(_2 F_1\) with the coefficients of each of the known approximations, for which maximum errors can then be established. These approximations range over four centuries from Kepler’s in 1609 to Almkvist’s in 1985 and include two from Ramanujan. Reviewer: Andrei Martínez Finkelshtein (Almeria) Cited in 1 ReviewCited in 42 Documents MSC: 33C20 Generalized hypergeometric series, \({}_pF_q\) 41A30 Approximation by other special function classes Keywords:hypergeometric; approximations; elliptic arc length Citations:Zbl 0948.30024 PDFBibTeX XMLCite \textit{R. W. Barnard} et al., SIAM J. Math. Anal. 32, No. 2, 403--419 (2000; Zbl 0983.33006) Full Text: DOI Digital Library of Mathematical Functions: §19.9(i) Complete Integrals ‣ §19.9 Inequalities ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals §19.9(i) Complete Integrals ‣ §19.9 Inequalities ‣ Legendre’s Integrals ‣ Chapter 19 Elliptic Integrals