Chapter 13 of Ramanujan's second notebook: integrals and asymptotic expansions
There are a number of gems in this chapter from Ramanujan's second notebook. One is an asymptotic expansion of \(_ 2F_ 1(1,m;m-n;(m- n)/n)\) when m, n and \(m-n>0\) tend to infinity. A second is two terms of the asymptotic expansion of \(\sum^{\infty}_{k=0}\prod^{k}_{j=1}\phi(\alpha h+j\delta h)/\phi(\beta h+j\gamma h)\) as \(h\to 0\) when \(\phi\) (x) is a function analytic and nonvanishing for \(| x| \leq d, \phi\) (x) and \(\phi\) '(x) are positive for \(x\geq -d\) and \(x\phi '(x)\geq M\phi(x)\) for \(x\geq d,\) where M is a positive constant. Many other interesting results, too numerous to mention here, were stated by Ramanujan, and are either proved in this paper, or a reference is given to a proof or statement in the literature.
scientific article; zbMATH DE number 3857484