Ramanujan type congruences for the Klingen-Eisenstein series

In the case of Siegel modular forms of degree \(n\) , we prove that, for almost all prime ideals \(\mathfrak {p}\) in any ring of algebraic integers, mod \(\mathfrak {p}^m\) cusp forms are congruent to true cusp forms of the same weight. As an application we give congruences for the Klingen-Eisenstein series and cusp forms, which can be regarded as a generalization of Ramanujan’s congruence. We will conclude by giving numerical examples.     

A new proof of the Ramanujan congruences for the partition function

Let p(n) denote the number of partitions of n. Recall Ramanujan’s three congruences for the partition function,

New Ramanujan type congruences modulo 5 for overpartitions

We present the transformation of several sums of positive integer powers of the sine and cosine into non-trigonometric combinatorial forms. The results are applied to the derivation of generating functions and to the number of the closed walks on a path and in a cycle.     

Generalization of five q-series identities of Ramanujan and unexplored weighted partition identities

The Ramanujan Journal - Ramanujan recorded five interesting q-series identities in a section that is not as systematically arranged as the other chapters of his second notebook. These five...     

Ramanujan type congruences for modular forms of several variables

We give congruences between the Eisenstein series and a cusp form in the cases of Siegel modular forms and Hermitian modular forms. We should emphasize that there is a relation between the existence of a prime dividing the (k?1)th generalized Bernoulli number and the existence of non-trivial Hermitian cusp forms of weight k. We will conclude by giving numerical examples for each case.     

There exist no Ramanujan congruences mod 6912

Let τ(n) be Ramanujan's function, $$x\prod _{m = 1}^\infty (1 - x^m )^{24} = \sum\nolimits_{n = 1}^\infty {\tau (n)x^n .} $$ In this paper it is shown that the Ramanujan congruence τ(n)=σ_d/nd¹¹ mod 691 cannot be improved mod 691². The following result is proved: for arbitrary r, s mod 691 the set of primes such that p ≡ r mod 691,τ (p) ≡ p¹¹+1+691 · s mod 691² has positive density.     

Convolution congruences for the partition function

Ahlgren and Boylan recently proved the uniqueness of the Ramanujan congruences for the primes , , and by using the modularity of a certain partition function. Here we use their result to find universal congruences, of a different type, which hold for the partition function modulo all primes .

     

On three identities of Ramanujan

The Ramanujan Journal - In this paper, we represent the generating function of the rank function as a summation of four parts—a constant, two Lambert series and a product. Applying it to...     

On Ramanujan congruences for modular forms of integral and half-integral weights

In 1916 Ramanujan observed a remarkable congruence: . The modern point of view is to interpret the Ramanujan congruence as a congruence between the Fourier coefficients of the unique normalized cusp form of weight and the Eisenstein series of the same weight modulo the numerator of the Bernoulli number . In this paper we give a simple proof of the Ramanujan congruence and its generalizations to forms of higher integral and half-integral weights.

     

Shift-plethystic trees and Rogers–Ramanujan identities

The Ramanujan Journal - By studying non-commutative series in an infinite alphabet, we introduce shift-plethystic trees and a class of integer compositions as new combinatorial models for the...     

Euler’s partition theorem for all moduli and new companions to Rogers–Ramanujan–Andrews–Gordon identities

The Ramanujan Journal - We generalise Euler’s partition theorem involving odd parts and distinct parts for all moduli and provide new companions to...     

On Ramanujan’s modular identities

For Ramanujan’s modular identities connected with his well-known partition congruences for the moduli 5 or 7, we had given, in an earlier paper, natural and uniform proofs through the medium of modular forms. Analogous (modular) identities corresponding to the (more difficult) case of the modulus 11 are provided here, with the consequent partition congruences; the relationship with relevant results of N J Fine is also sketched.     

Some new identities for colored partition

In this paper, we discover several new identities on colored partitions and provide proofs for them. Many colored partition identities presented in the paper do not belong to the general and unified combinatorial framework provided by Sandon and Zanello. Most of our proofs depend upon new modular equations given by employing the method of reciprocation.     

Capelli identities for reductive dual pairs

Analogues of the Capelli identity are given for every irreducible reductive dual pair in the complex symplectic group and the complex orthogonal group. They describe a natural correspondence between the “centers” of the two universal enveloping algebras and an algebra of invariant operators.     

Vanishing coefficients and identities concerning Ramanujan’s parameters

The Ramanujan Journal - In this paper we investigate several infinite products with vanishing Taylor coefficients in arithmetic progressions. These infinite products are closely related to...     

Parity in partition identities

This paper considers a variety of parity questions connected with classical partition identities of Euler, Rogers, Ramanujan and Gordon. We begin by restricting the partitions in the Rogers-Ramanujan-Gordon identities to those wherein even parts appear an even number of times. We then take up questions involving sequences of alternating parity in the parts of partitions. This latter study leads to: (1) a bi-basic q-binomial theorem and q-binomial series, (2) a new interpretation of the Rogers-Ramanujan identities, and (3) a new natural interpretation of the fifth-order mock theta functions f ₀(q) along with a new proof of the Hecke-type series representation.     

Rogers–Ramanujan type identities via Abel’s lemma on summation by parts

The Ramanujan Journal - The Abel’s lemma on summation by parts is employed to review identities of Rogers–Ramanujan type. Twenty examples are illustrated including several new RR...