On Ramanujan asymptotic expansions and inequalities for hypergeometric functions

Abstract

In this paper we first discuss refinement of the Ramunujan asymptotic expansion for the classical hypergeometric functions F(a,b;c;x), c ≤ a + b, near the singularity x = 1. Further, we obtain monotonous properties of the quotient of two hypergeometric functions and inequalities for certain combinations of them. Finally, we also solve an open problem of finding conditions on a, b > 0 such that 2F(−a,b;a + b;r²) < (2−r²)F(a,b;a + b;r²) holds for all r∈(0,1).