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Asymptotic formulas for zero-balanced hypergeometric series. (English) Zbl 0547.33001
An asymptotic formula for the partial sums of a Clausen function of unit argument in which the sum of the denominator parameters is the same as that of the numerator parameters. This latter condition is of particular interest in view of the fact that a non-terminating Clausen function of unit argument converges only if the sum of the denominator parameters exceeds the sum of the numerator parameters, so that, in this paper, a useful limiting case is discussed. Some basic analogues of the main results of this paper are also given. It is recalled that the Indian mathematician Ramanujan in his notebooks (S. Ramanujan, Notebooks, 2 volumes, TATA Institute of Fundamental Research, Bombay, India) has stated without proof some of the expressions deduced here.
Reviewer: H.Exton

MSC:
 33C05 Classical hypergeometric functions, $${}_2F_1$$ 33E05 Elliptic functions and integrals
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