A relationship between certain colored generalized Frobenius partitions and ordinary partitions

AbstractRamanujan's congruence p(5n + 4) ≡ 0 (mod 5) for ordinary partitions is well-known. This congruence is just the first in a family of congruences modulo 5; namely, p(5nn + δα) ≡ 0 (mod 5α) for α ≧ 1 where δα represents the reciprocal of 24 modulo 5α. A similar family of congruences exists for ordinary partitions modulo 7. In this paper we prove the corresponding congruences for generalized Frobenius partitions with 5 and 7 colors modulo 5 and 7, respectively, by establishing an equality between these two classes of generalized Frobenius partitions and certain ordinary partitions. The proofs are based on some elegant identities of Ramanujan

Bilateral identities of the Rogers-Ramanujan type

We derive by analytic means a number of bilateral identities of the Rogers-Ramanujan type. Our results include bilateral extensions of the Rogers-Ramanujan and the G\"ollnitz-Gordon identities, and of related identities by Ramanujan, Jackson, and Slater. We give corresponding results for multiseries including multilateral extensions of the Andrews-Gordon identities, of Bressoud's even modulus identities, and other identities. The here revealed closed form bilateral and multilateral summations appear to be the very first of their kind. Given that the classical Rogers-Ramanujan identities have well-established connections to various areas in mathematics and in physics, it is natural to expect that the new bilateral and multilateral identities can be similarly connected to those areas. This is supported by concrete combinatorial interpretations for a collection of four bilateral companions to the classical Rogers-Ramanujan identities.Comment: 25 page

On the density of the odd values of the partition function

The purpose of this note is to introduce a new approach to the study of one of the most basic and seemingly intractable problems in partition theory, namely the conjecture that the partition function $p(n)$ is equidistributed modulo 2. Our main result will relate the densities, say $\delta_t$, of the odd values of the $t$-multipartition functions $p_t(n)$, for several integers $t$. In particular, we will show that if $\delta_t>0$ for some $t\in \{5,7,11,13,17,19,23,25\}$, then (assuming it exists) $\delta_1>0$; that is, $p(n)$ itself is odd with positive density. Notice that, currently, the best unconditional result does not even imply that $p(n)$ is odd for $\sqrt{x}$ values of $n\le x$. In general, we conjecture that $\delta_t=1/2$ for all $t$ odd, i.e., that similarly to the case of $p(n)$, all multipartition functions are in fact equidistributed modulo 2. Our arguments will employ a number of algebraic and analytic methods, ranging from an investigation modulo 2 of some classical Ramanujan identities and several other eta product results, to a unified approach that studies the parity of the Fourier coefficients of a broad class of modular form identities recently introduced by Radu.Comment: Several changes with respect to the 2015 version. 18 pages. To appear in the Annals of Combinatoric