On Modular Relations for the Göllnitz–Gordon Functions with Applications to Partitions

In this paper, we research theta function identity involving Rogers– Ramanujan identity and establish a Rogers–Ramanujan–Slater type theta function identity related to G(q) and φ(q).

In his first and second letters to Hardy, Ramanujan made several assertions about the Rogers-Ramanujan continued fraction F ( q ) F(q) . In order to prove some of these claims, G. N. Watson established two important theorems about F ( q ) F(q) that he found in Ramanujan’s notebooks. In his lost notebook, after stating a version of the quintuple product identity, Ramanujan offers three theta function identities, two of which contain as special cases the celebrated two theorems of Ramanujan proved by Watson. Using addition formulas, the quintuple product identity, and a new general product formula for theta functions, we prove these three identities of Ramanujan from his lost notebooks.

Modular relations involving generalized digamma functions

Chen and Huang established some elegant modular relations for the Gollnitz–Gordon functions analogous to Ramanujan’s list of forty identities for the Rogers–Ramanujan functions. In this paper, we derive some new modular relations involving cubes of the Gollnitz–Gordon functions. Furthermore, we also provide new proofs of some modular relations for the Gollnitz–Gordon functions due to Gugg.

Ramanujan's “Lost” Notebook VIII: the entire Rogers–Ramanujan function

Basis partition polynomials, overpartitions and the Rogers–Ramanujan identities

We prove a theorem which adds a new companion to the Rogers–Ramanujan identities. This new companion counts partitions with different type of constraints on even and odd parts. Generalizing this theorem, we obtain two families of partition identities.

In a recent paper, Frechette and Locus examined and found expressions for the infinite product [math] in terms of products of [math] -series of the Rogers–Ramanujan type coming from Hall–Littlewood polynomials, when [math] . These [math] -series were originally discovered in 2014 by Griffin, Ono, and Warnaar in their work on the framework of the Rogers–Ramanujan identities. Extending this framework, Rains and Warnaar also recently discovered more [math] -series and their corresponding infinite products. Frechette and Locus left open the case where [math] . Here we find such an expression for the infinite products for [math] by making use of the new [math] -series obtained by Rains and Warnaar.

We give an explicit recursive description of the Hilbert series and Grobner bases for the family of quadratic ideals defining the jet schemes of a double point. We relate these recursions to the Rogers–Ramanujan identity and prove a conjecture of the second author, Oblomkov and Rasmussen.

Modular equations for cubes of the Rogers–Ramanujan and Ramanujan–Göllnitz–Gordon functions and their associated continued fractions

Partial-Sum Analogues of the Rogers–Ramanujan Identities

On the Number of Homomorphisms from a Finite Group to a General Linear Group

Finite Affine Groups: Cycle Indices, Hall–Littlewood Polynomials, and Probabilistic Algorithms

The method of combinatorial telescoping

Corteel, Lovejoy, and Mallet concluded their paper “An extension to overpartitions of the Rogers–Ramanujan identities for even moduli” with an open question of investigating the combinatorial properties of a q-series with two additional parameters. We settle their question, unfortunately in the negative, by showing that the series yields only the known results in overpartitions. However, when one annihilates one of the parameters, the resulting series have nice integer partitions interpretations. Those series appeared in another publication as well. In particular, Corteel, Lovejoy, and Mallet’s series involve an index d. This index unifies two classes of overpartition identities for \(d = 1\) and \(d = 2\), but does not give additional overpartition identities for \(d \ge 3\). Upon setting one of the parameters zero, one does get regular partition identities for all d. The proofs are conventional, formal verifications for brevity, but we show how to make the proofs constructive.