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Zbl 0539.33001
Berndt, Bruce C.; Evans, Ronald J.
Chapter 13 of Ramanujan's second notebook: integrals and asymptotic expansions.
(English)
[J] Expo. Math. 2, 289-347 (1984). ISSN 0723-0869

There are a number of gems in this chapter from Ramanujan's second notebook. One is an asymptotic expansion of $\sb 2F\sb 1(1,m;m-n;(m- n)/n)$ when m, n and $m-n>0$ tend to infinity. A second is two terms of the asymptotic expansion of $\sum\sp{\infty}\sb{k=0}\prod\sp{k}\sb{j=1}\phi(\alpha h+j\delta h)/\phi(\beta h+j\gamma h)$ as $h\to 0$ when $\phi$ (x) is a function analytic and nonvanishing for $\vert x\vert \le d, \phi$ (x) and $\phi$ '(x) are positive for $x\ge -d$ and $x\phi '(x)\ge M\phi(x)$ for $x\ge d,$ 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.
MSC 2000:
*33-02 Research monographs (special functions)
33C05 Classical hypergeometric functions
41A60 Asymptotic problems in approximation
33E99 Special functions

Keywords: confluent hypergeometric function; gamma function; beta integrals; Ramanujan's second notebook

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