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Chapter 13 of Ramanujan’s second notebook: integrals and asymptotic expansions. (English) Zbl 0539.33001
There are a number of gems in this chapter from Ramanujan’s second notebook. One is an asymptotic expansion of 2 F 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 k=0 j=1 k φ(αh+jδh)/φ(βh+jγh) as h0 when φ (x) is a function analytic and nonvanishing for |x|d,φ (x) and φ ’(x) are positive for x-d and xφ ' (x)Mφ(x) for xd, 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.
Reviewer: R.Askey
33-02Research monographs (special functions)
33C05Classical hypergeometric functions, 2 F 1
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
33E99Other special functions