# Further Identities and Congruences for the Coefficients of Modular Forms

@article{Newman1958FurtherIA, title={Further Identities and Congruences for the Coefficients of Modular Forms}, author={Morris Newman}, journal={Canadian Journal of Mathematics}, year={1958}, volume={10}, pages={577 - 586} }

If n is a non-negative integer, define p r(n) by otherwise define p r(n) as 0. (Here and in what follows all sums will be extended from 0 to ∞ and all products from 1 to ∞ unless otherwise stated.) p r(n) is thus generated by the powers of ,

## 17 Citations

Distribution of a certain partition function modulo powers of primes

- Mathematics
- 2011

In this paper, we study a certain partition function a(n) defined by Σn≥0a(n)qn:= Πn=1(1 − qn)−1(1 − q2n)−1. We prove that given a positive integer j ≥ 1 and a prime m ≥ 5, there are infinitely many…

Some Powers of The Euler Product

- Mathematics
- 1990

The coefficients of the r th power of the Euler product for r even, 0 < r ≤ 24, are a natural generalization of the Ramanujan r-function, and satisfy similar recurrence formulas. These coefficients…

Note on Partitions Modulo 5

- Mathematics
- 1967

In the paper preceding this one, Parkin and Shanks study the distribution of the values of the unrestricted partition function p(n) modulo 2 and come to the conclusion that there is no apparent…

Applications of the theory of modular forms to number theory

- Mathematics
- 1980

The survey is devoted to arithmetic questions in the theory of modular forms and, in particular, to arithmetic applications of modular functions; mainly only elementary and analytic aspects of this…

SOME PROPERTIES OF p(n) AND c(n) MODULO POWERS OF 13

- Mathematics
- 1967

j(^r) = E c(n)xn = (1 +240 0 u3(n)xn) /xf24(x) 744, where 03(n) = d3. dln Then p(n) is just the number of unrestricted partitions of n, and c(n) is the Fourier coefficient of Klein's modular…

A REFINEMENT OF RAMANUJAN'S CONGRUENCES MODULO POWERS OF 7 AND 11

- Mathematics
- 2012

Ramanujan's famous congruences for the partition function modulo powers of 5, 7, and 11 have inspired much further research. For example, in 2002 Lovejoy and Ono found subprogressions of 5jn + β5(j)…

SOME CONGRUENCES DEDUCIBLE FROM RAMANUJAN'S CUBIC CONTINUED FRACTION

- Mathematics
- 2011

We present some interesting Ramanujan-type congruences for some partition functions arising from Ramanujan's cubic continued fraction. One of our results states that if p3(n) is defined by $\sum_{n =…

Proof of a conjecture of Ramanujan

- MathematicsGlasgow Mathematical Journal
- 1967

We write and so that p(n) is the number of unrestricted partitions of n. Ramanujan [1] conjectured in 1919 that if q = 5, 7, or 11, and 24m ≡ 1 (mod qn), then p(m) ≡ 0 (mod qn). He proved his…

Exceptional Congruences for Powers of the Partition Function

- Mathematics
- 2004

AbstractIn Journal of London Math. Soc.31 (1956), 350–359, Morris Newman studied vector spaces of functions arising from lifts to Γ0(p) of certain eta-products on the group Γ0(pQ), Q = pn. In this…

## References

SHOWING 1-4 OF 4 REFERENCES

Identities analogous to Ramanujaris identities involving the partition function

- Duke Math. J.,
- 1939

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

- MathematicsCanadian Journal of Mathematics
- 1957

If n is a non-negative integer, define p r(n) as the coefficient of x n in ; otherwise define p r(n) as 0. In a recent paper (2) the author established the following congruence: Let r = 4, 6, 8, 10,…