Related papers: Rogers-Ramanujan identities in Statistical Mechani…
Andrews investigated parity conditions in the Rogers-Ramanujan-Gordon theorem. Under the conditions that even parts or odd parts appear an even number of times, Andrews discovered two Rogers-Ramanujan-Gordon type partition theorems and…
In 1915, Ramanujan proved asymptotic inequalities for the sum of divisors function, assuming the Riemann hypothesis (RH). We consider a strong version of Ramanujan's theorem and define highest abundant numbers that are extreme with respect…
We give a commutative algebra viewpoint on Andrews recursive formula for the partitions appearing in "Gordon's identities", which are a generalization of Rogers-Ramanujan identities. Using this approach and differential ideals we conjecture…
In this paper, we first establish two new Bailey pairs via finding two generalizations of Euler's pentagonal number theorem. Next, we specificize the Bailey lemmas with these two Bailey pairs. As applications, we finally establish some…
We give an explicit recursive description of the Hilbert series and Gr\"obner 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…
In the past 30 years there have been extensive discoveries in the theory of integrable statistical mechanical models including the discovery of non-linear differential equations for Ising model correlation functions, the theory of random…
Relations involving the Rogers-Ramanujan continued fractions $R(q),$ $R(q^3 ),$ and $R(q^4)$ are used to find new generating functions and congruences modulo 5 and 25 for 3-core, 4-core, 4-regular, and colored partition functions.
Quite recently, the first author investigated vanishing coefficients of the arithmetic progressions in several $q$-series expansions. In this paper, we further study the signs of coefficients in two $q$-series expansions and establish some…
Using new $q$-functions recently introduced by Hatayama et al. and by (two of) the authors, we obtain an A_2 version of the classical Bailey lemma. We apply our result, which is distinct from the A_2 Bailey lemma of Milne and Lilly, to…
A new type of polynomial analogue of the Rogers-Ramanujan identities is proven. Here the product-side of the Rogers-Ramanujan identities is replaced by a partial theta sum and the sum-side by a weighted sum over Schur polynomials.
Kanade and Russell conjectured several Rogers-Ramanujan-type partition identities, some of which are related to level $2$ characters of the affine Lie algebra $A_9^{(2)}$. Many of these conjectures have been proved by Bringmann,…
In this article we continue a previous work in which we have generalized the Rogers Ramanujan continued fraction (RR) introducing what we call, the Ramanujan-Quantities (RQ). We use the Mathematica package to give several modular equations…
Let $R(q)$ denote the Rogers-Ramanujan continued fraction for $|q| < 1$. By applying the RootApproximant command in the Wolfram language to expressions involving the theta function $f(-q) := (q;q)_{\infty}$ given in modular relations due to…
We study the generating functions for cylindric partitions with profile $(c_1,c_2,c_3)$ for all $c_1,c_2,c_3$ such that $c_1+c_2+c_3=5$. This allows us to discover and prove seven new $A_2$ Rogers-Ramanujan identities modulo $8$ with…
Polynomial generalizations of all 130 of the identities in Slater's list of identities of the Rogers-Ramanujan type are presented. Furthermore, duality relationships among many of the identities are derived. Some of the these polynomial…
Arc spaces have been introduced in algebraic geometry as a tool to study singularities but they show strong connections with combinatorics as well. Exploiting these relations we obtain a new approach to the classical Rogers-Ramanujan…
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 identities do not seem to have acquired enough attention. Recently, Dixit…
In 1961, Baker, Gammel and Wills conjectured that for functions $f$ meromorphic in the unit ball, a subsequence of its diagonal Pad\'{e} approximants converges uniformly in compact subsets of the ball omitting poles of $f$. There is also…
A generalization of a beautiful $q$-series identity found in the unorganized portion of Ramanujan's second and third notebooks is obtained. As a consequence, we derive a new three-parameter identity which is a rich source of…
We conjecture polynomial identities which imply Rogers--Ramanujan type identities for branching functions associated with the cosets $({\cal G}^{(1)})_{\ell-1}\otimes ({\cal G}^{(1)})_{1} / ({\cal G}^{(1)})_{\ell}$, with ${\cal…