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Identities and Congruences of the Ramanujan Type

  • K. G. Ramanathan (a1)

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Let P(n) denote the number of unrestricted partitions of the positive integer n. Ramanujan conjectured that

(1.1)

if . He also indicated that such congruences could be deduced from identities of the type

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References

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1 Ramanujan, S. Collected papers, Cambridge, 1927.

2 Watson, G. N., Ramanujans Vermutung Über Zerfällungsanzahlen, Jour, für Math., Bd. 179,(1938), p. 97.

3 H., Rademacher The Ramanujan identities under modular substitutions. Trans. Amer. Math. Soc, vol. 51, (1942) p. 609.

4 A11 these are contained in Klein and Fricke, Vorlesungen iiber die Theorie der Elliptischen Modulfunktionen. Bd. 2, p. 64. Also see: Mordell, L. J., Note on certain modular relations considered by Messrs. Ramanujan, Darling, and Rogers. Proc. Lond. Math. Soc. (2), 20 (1922) p. 408.

5 depends on p. We omit this suffix p in general, but when explicit reference has to be made we write .

6 Watson, loc. cit., pp. 106, 119, 120.

7 There is another method of obtaining expressions for as polynomials in . This will be published elsewhere.

8 shall mean summation from i = 1 to a sufficiently large i.

9 The construction of the function Fn,v(T) is suggested by the work of Rademacher, p. 622 and 624.

10 To avoid complication we have not shown in ai(n), its dependence on v and p.

The author would like to thank Dr. Paul T. Bateman for his suggestions for improvement.

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Identities and Congruences of the Ramanujan Type

  • K. G. Ramanathan (a1)

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