Arc spaces and the Rogers–Ramanujan identities

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 Identities. The linking object is the Hilbert–Poincaré series of the arc space over a point of the base variety. In the case of the double point, this is precisely the generating series for the integer partitions without equal or consecutive parts.     

Topological strings,quiver varieties,and Rogers–Ramanujan identities

The Ramanujan Journal - Motivated by some recent works on BPS invariants of open strings/knot invariants, we guess there may be a general correspondence between the Ooguri–Vafa invariants of...     

Bressoud polynomials,Rogers–Ramanujan type identities,and applications

In his paper providing an easy proof of the Rogers–Ramanujan identities, D. Bressoud extended his work to multiple series identities. Intrinsic in his works are polynomials with diverse applications to several aspects of \(q\)-series. This paper provides an initial exploration of these polynomials.     

Polygonal numbers and Rogers–Ramanujan–Gordon theorem

The Ramanujan Journal - Let $$B_{k,r}(n)$$ be the number of partitions of the form $$n=b_1+b_2+cdots +b_s$$ , where $$b_i-b_{i+k-1}hbox {,char 062,}2$$ and at most $$r-1$$ of the $$b_i$$ are...     

Weighted Rogers–Ramanujan partitions and Dyson crank

In this paper, we refine a weighted partition identity of Alladi. We write formulas for generating functions for the number of partitions grouped with respect to a partition statistic other than the norm. We tie our weighted results as well as the different statistics with the crank of a partition. In particular, we prove that the number of partitions into even number of distinct parts whose odd-indexed parts’ sum is n is equal to the number of partitions of n with non-negative crank.     

On the Andrews–Schur proof of the Rogers–Ramanujan identities

In this article, we study one of Andrews’ proofs of the Rogers–Ramanujan identities published in 1970. His proof inspires connections to some famous formulas discovered by Ramanujan. During the course of study, we discovered identities such as $$\sum_{n\geq0}\frac{q^{n^2}}{(q;q)_n}=\frac{1}{\sqrt{5}}\Biggl(\beta \prod_{n=1}^{\infty}\frac{1}{1+\alpha q^{n/5}+q^{2n/5}}-\alpha \prod_{n=1}^{\infty}\frac{1}{1+\beta q^{n/5}+q^{2n/5}}\Biggr),$$ where β=?1/α is the Golden Ratio.     

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...     

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...     

Odd values of the Rogers–Ramanujan functions

Let $g(n)$ and $h(n)$ be the coefficients of the Rogers–Ramanujan identities. We obtain asymptotic formulas for the number of odd values of $g(n)$ for odd n, and $h(n)$ for even n, which improve Gordon's results. We also obtain lower bounds for the number of odd values of $g(n)$ for even n, and $h(n)$ for odd n.     

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

In this paper, we prove modular identities involving cubes of the Rogers–Ramanujan functions. Applications are given to proving relations for the Rogers–Ramanujan continued fraction. Some of our identities are new. We establish analogous results for the Ramanujan–Göllnitz–Gordon functions and the Ramanujan–Göllnitz–Gordon continued fraction. Finally, we offer applications to the theory of partitions.     

The Rogers–Selberg recursions, the Gordon–Andrews identities and intertwining operators

Using the theory of intertwining operators for vertex operator algebras we show that the graded dimensions of the principal subspaces associated to the standard modules for satisfy certain classical recursion formulas of Rogers and Selberg. These recursions were exploited by Andrews in connection with Gordon’s generalization of the Rogers–Ramanujan identities and with Andrews’ related identities. The present work generalizes the authors’ previous work on intertwining operators and the Rogers–Ramanujan recursion. 2000 Mathematics Subject Classification Primary—17B69, 39A13 S. Capparelli gratefully acknowledges partial support from MIUR (Ministero dell’Istruzione, dell’Università e della Ricerca). J. Lepowsky and A. Milas gratefully acknowledge partial support from NSF grant DMS-0070800.     

On the 2- and 4-dissections of the Rogers–Ramanujan functions

Basil Gordon showed that if the two Rogers–Ramanujan functions are expanded as series, in each case half the coefficients are almost always even. In order to do this, he gave formulae modulo 2 for the 2-dissections of each of the Rogers–Ramanujan functions. In this paper we give the exact 2- and 4-dissections. We then show how our 2-dissections lead to Gordon’s formulae, we go on to give examples of what he calls “linear zero congruences modulo 2” , and finally we give a simple proof of Gordon’s main result that \(g(2n+1)\) and \(h(2n)\) are almost always even.     

A simple proof of the Rogers–Selberg identities

A short and simple proof of the Rogers–Selberg identities is obtained from the work of Andrew Sills on identities of the Rogers–Ramanujan type.     

Born–Infeld solitons,maximal surfaces,and Ramanujan’s identities

We show that a Born–Infeld soliton can be realised either as a spacelike minimal graph or timelike minimal graph over a timelike plane or a combination of both away from singular points. We also obtain some exact solutions of the Born–Infeld equation from already known solutions to the maximal surface equation. Further we present a method to construct a one parameter family of complex solitons from a given one parameter family of maximal surfaces. Finally, using Ramanujan’s identities and the Weierstrass–Enneper representation of maximal surfaces, we derive further non-trivial identities.     

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...     

On Plouffe’s Ramanujan identities

Recently, Simon Plouffe has discovered a number of identities for the Riemann zeta function at odd integer values. These identities are obtained numerically and are inspired by a prototypical series for Apéry’s constant given by Ramanujan:

$\zeta(3)=\frac{7\pi^{3}}{180}-2\sum_{n=1}^{\infty}\frac {1}{n^{3}(e^{2\pi n}-1)}.$

Such sums follow from a general relation given by Ramanujan, which is rediscovered and proved here using complex analytic techniques. The general relation is used to derive many of Plouffe’s identities as corollaries. The resemblance of the general relation to the structure of theta functions and modular forms is briefly sketched.