The Ramanujan identities under modular substitutions

@article{Rademacher1942TheRI,
  title={The Ramanujan identities under modular substitutions},
  author={H. A. Rademacher},
  journal={Transactions of the American Mathematical Society},
  year={1942},
  volume={51},
  pages={609-636}
}
  • H. Rademacher
  • Published 1942
  • Mathematics
  • Transactions of the American Mathematical Society
they appear in a form which suggests certain group-theoretical considerations, similar to those employed by Hecke in his theory of modular forms. In this way we transform the identities into new ones which are noteworthy because of the occurrence of the Legendre symbol and which, by a simple further argument, lead also to a proof of (1.1) and (1.2). An analogous identity for the modulus 13, given by Zuckerman, can be treated in the same way. G. N. Watson and H. S. Zuckerman have also derived… Expand
Remarks on some modular identities
Introduction. We shall consider a certain class of functions invariant with respect to the substitutions of the congruence subgroup Fo(p) of the modular group r. By specializing these functions, weExpand
Divisibility Properties of the Fourier Coefficients of the Modular Invariant j(τ)
The general idea behind the proof of this theorem is as follows. Consider the congruence (1. 2) as an example. By applying a certain linear operatordenoted by U5-to the right member of (1. 1) oneExpand
On the Minimal Level of Modular Forms
In this paper, we will deal with a very practical problem. The problem is that of finding the lowest level of a given modular form which is known to have some level. For forms on principalExpand
An Implementation of Radu's Ramanujan-Kolberg Algorithm
In 2015 Cristian-Silviu Radu designed an algorithm to detect identities of a class studied by Ramanujan and Kolberg. This class includes the famous identities by Ramanujan which provide a witness toExpand
Ramanujan's circular summation, t-cores and twisted partition identities
Abstract In this paper, we give new evaluations for Ramanujan's circular summation function. We also provide simpler proofs for known evaluations and give some generalizations. We discover modularExpand
On a conjecture of Atkin
Abstract Let j be the modular invariant. For the primes p ≦ 23 the q-expansion coefficients of Um (j – 744) are multiplicative as it was a Hecke eigenform modulo a power of p which increases with m.Expand
Ramanujan’s Unpublished Manuscript on the Partition and Tau Functions with Proofs and Commentary
When Ramanujan died in 1920, he left behind an incomplete, unpublished manuscript in two parts on the partition function p(n) and, in contemporary terminology, Ramanujan’s tau-function τ(n). TheExpand
Ramanujan’s Unpublished Manuscript on the Partition and Tau Functions
When Ramanujan died in 1920, he left behind an incomplete, unpublished manuscript in two parts on the partition function p(n) and, in contemporary terminology, Ramanujan’s tau-function τ(n). TheExpand
Dedekind $\eta$-function, Hauptmodul and invariant theory
We solve a long-standing open problem with its own long history dating back to the celebrated works of Klein and Ramanujan. This problem concerns the invariant decomposition formulas of theExpand
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 thisExpand
...
1
2
3
4
...

References

SHOWING 1-8 OF 8 REFERENCES
Identities analogous to Ramanujan’s identities involving the parition function
Ramanujans Vermutung über Zerfällungszahlen.
Mordell, Note on certain modular relations considered by Messrs
  • Proceedings of the London Mathematical Society,
  • 1922
und R
  • Fricke, Vorlesungen über die Theorie der Elliptischen Modulfunktionen, vol. 1, 1890, and vol. 2,
  • 1892