Analytic Number Theory: Proceedings of a Conference Held at by Emil Grosswald (auth.), Marvin I. Knopp (eds.)

By Emil Grosswald (auth.), Marvin I. Knopp (eds.)

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Additional resources for Analytic Number Theory: Proceedings of a Conference Held at Temple University, Philadelphia, May 12–15, 1980

Example text

3), we have = ~ Cn(p)(-ix)n p2+2pcos x + l n=O n~(p+l) n+l Equating real and imaginary parts on both sides above, we deduce Corollaries 4 and 5, respectively. In the hypothesis of Corollary 6, which consists of four parts, Ramanujan attempts to define a sequence of numbers {An}. uniquely determined. However, these numbers are not I t is preferable to define An by Corollary 6(iv) and then deduce the equality of the hypothesis. Hence, we have taken the liberty of inverting (iv) and the hypothesis below.

31 1737 7290 62 32 1940 8267 -29 33 2153 9332 -71 34 2389 10500 41 35 2648 11776 81 6. Conclusion. Our object in this paper has been to show the relationship of one particular set of identities from Ramanujan's "Lost" Notebook to the Mordell integrals. N. Watson [24] did for the third order mock theta functions, Indeed we have primarily restricted our attention to the case a = -I when s p e c i a l i z i n g the i d e n t i t i e s of Section 3. Howeveri t is clear that i n t e r e s t i n g real results arise whenever a is an algebraic u n i t and at most a quadratic i r r a t i o n a l i t y .

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. I, 49-239 of Jacobi's, Gesammelte Werke, Reimer, Berlin 1881-1891, now by Chelsea, New York, 1969). 15. N. , 13 (1974), 383-436. 16. J. Mordell, The value of the definite integral f " eat2+btdt/(eCt+d), Quarterly Journal of Math,, 68 C1920), 329-342.

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