Abstract:
Conditions are determined under which 3F2 (−n, a, b; a + b + 2, ε − n + 1; 1) is a
monotone function of n satisfying ab· 3F2 (−n, a, b; a + b + 2, ε − n + 1; 1) ≥ ab· 2F1 (a, b; a + b + 2; 1) .
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 3F2(−n, − 1
2 , − 1
2 ; 1, ε − n + 1; 1) ≥ 4
π , for 1 > ≥ 1
4 and n ≥ 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.
