A Monotonicity Property Involving ₃<i>F</i>₂ and Comparisons of the Classical Approximations of Elliptical Arc Length

Abstract

Conditions are determined under which $ \,_{3}F_{2}\left(-n,a,b;a+b+2,\varepsilon -n+1;1\right)$ is a monotone function of n satisfying $a b\cdot\,_{3}F_{2}\left(-n,a,b;a+b+2,\varepsilon -n+1;1\right) \geq a b\cdot\,_{2}F_{1}\left(a,b;a+b+2;1\right).$ Motivated by a conjecture of Vuorinen [Proceedings of Special Functions and Differential Equations, K. S. Rao, R. Jagannathan, G. Vanden Berghe, J. Van der Jeugt, eds., Allied Publishers, New Delhi, 1998], the corollary that $\,_{3}F_{2}(-n,-\frac{1}{2},-\frac{1}{2};1,\varepsilon -n+1;1) \geq \frac{4}{\pi},$ for $1>\epsilon \geq\frac{1}{4}$ 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 2F1 , 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.