# 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},
year={1958},
volume={10},
pages={577 - 586}
}
• M. Newman
• Published 1958
• Mathematics
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 ,
Distribution of a certain partition function modulo powers of primes
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
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
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
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
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
• A. Atkin
• Mathematics
Glasgow Mathematical Journal
• 1967
We write and so that p(n) is the number of unrestricted partitions of n. Ramanujan  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
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