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Extra info for Analytic Number Theory: Proceedings of a Conference Held at Temple University, Philadelphia, May 12–15, 1980
17. J. Mordell, The definite integral f " eax2+bXdx/(eCX+d) and the analytic theory of numbers, Acta Math. Stockholm, 61 (1933), 323-360. 18. S. Ramanujan, Collected Papers, Cambridge University Press, London and New York, 1927; reprinted Chelsea, New York. 19. B. Sears, On the transformation theory of basic hypergeometric functions, Proc. London Math. Soc. (2), 53 (1951), 158-180. 20. L. Siegel, Uber Riemanns Nachlass zur analytischen Zahlentheorie, Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, 2 (1933), 45-80.
N. Bailey, On the basic bilateral hypergeometric series 2~2, Quart. J. , l (1950), 194-198. II. R. Bellman, A Brief Introduction to Theta Functions, Holt, Rinehart and Winston, New York, 1961. 12. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, Oxford University Press, London, 1935. 13. A. Dragonette, Some asymptotic formulae for the mock theta series of Ramanujan, Amer. J. , 72 (1952), 474-500. 14. J. Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum, Regiomontl, fratrum Borntrager, 1829 (Reprinted: in vol.
Extensions and modular transformations of Ramanujan's functions. While the approach of the last section provides uniformity, it fails to place these results in the basic hypergeometric hierarchy. In fact many of the results of Section 3 may be viewed as specializations of Watson's q-analog of Whipple's theorem [21; p. I00, eq. 1) 8r I , _~, ~ d, ~q ' c e, ~q ' d q-N ; q, ~2qN+2. , ~q deq-N ar; q ' t i b l , b 2 . . . 4). 7) when a = I. Most noteworthy in a l l these observations however is the f a c t t h a t n e i t h e r M3(q) nor a g e n e r a l i z a t i o n appears in any of Ramanujan's work.