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Congruences for the Coefficients of Modular forms and Some New Congruences for the Partition Function

Published online by Cambridge University Press:  20 November 2018

Morris Newman
Morris Newman*
Affiliation:
National Bureau of Standards Washington, D.C.
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If n is a non-negative integer, define pr(n) as the coefficient of xn in

;

otherwise define pr(n) as 0. In a recent paper (2) the author established the following congruence:

Let r = 4, 6, 8, 10, 14, 26. Let p be a prime greater than 3 such that r(p + l) / 24 is an integer, and set Δ = r(p2 − l)/24.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 9 , 1957 , pp. 549 - 552
Copyright
Copyright © Canadian Mathematical Society 1957

References

1. Gupta, H., On a conjecture of Ramanujan, Proc. Ind. Acad. Sciences A, 4, (1936), 625–629.Google Scholar
2. Newman, M., Some Theorems about pr(n), Can. J. Math., 9 (1957). 68–70.Google Scholar
3. Newman, M., The coefficients of certain infinite products, Proc. Amer. Math. Soc, 4, (1953), 435–439.Google Scholar
4. Newman, M., Remarks on some modular identities, Trans. Amer. Math. Soc, 73 (1952), 313–320.Google Scholar
5. Newman, M., A table of the coefficients of the powers of n(τ), Proc. Kon. Nederl. Akad. Wetensch. Ser. A57 = Indagationes Math., 18 (1956), 204–216.Google Scholar
6. Watson, G. N., A table of Ramanujan1 s function τ(n), Proc. London Math. Soc, 51 (1949), 1–13.Google Scholar
7. Zuckerman, H., Identities analagous to Ramanujan's identities involving the partition function, Duke Math. J., 5 (1939), 88–110.Google Scholar