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A monotonicity property involving ${}_{3}{F}_{2}$ and comparisons of the classical approximations of elliptical arc length. (English) Zbl 0983.33006
Authors’ abstract: Conditions are determined under which ${}_{3}{F}_{2}\left(-n,a,b;a+b+2,\epsilon -n+1;1\right)$ is a monotone function of $n$ satisfying ${}_{3}{F}_{2}\left(-n,a,b;a+b+2,\epsilon -n+1;1\right)\ge a{b}_{2}{F}_{1}\left(a,b;a+b+2;1\right)$. 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 ${}_{3}{F}_{2}\left(-n,-1/2,-1/2;1,\epsilon -n+1;1\right)\ge 4/\pi$, for $1>\epsilon >1/4$ and $n\ge 2$, 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.

##### MSC:
 33C20 Generalized hypergeometric series, ${}_{p}{F}_{q}$ 41A30 Approximation by other special function classes
##### Keywords:
hypergeometric; approximations; elliptic arc length