Related papers: Rogers-Ramanujan type identities and Nil-DAHA
We prove Rogers-Ramanujan type identities for the Nahm sums associated with the tadpole Cartan matrix of rank $3$. These identities reveal the modularity of these sums, and thereby we confirm a conjecture of Penn, Calinescu and the first…
The dilute A$_3$ lattice model in regime 2 is in the universality class of the Ising model in a magnetic field. Here we establish directly the existence of an E$_8$ structure in the dilute A$_3$ model in this regime by expressing the…
This is a tutorial for using two new MAPLE packages, thetaids and ramarobinsids. The thetaids package is designed for proving generalized eta-product identities using the valence formula for modular functions. We show how this package can…
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 Rogers-Ramanujan identities. We prove that the language associated to…
We prove some new modular identities for the Rogers\textendash Ramanujan continued fraction. For example, if $R(q)$ denotes the Rogers\textendash Ramanujan continued fraction, then…
We present several new families of Rogers-Ramanujan type identities related to the moduli 18 and 24. A few of the identities were found by either Ramanujan, Slater, or Dyson, but most are believed to be new. For one of these families, we…
In this paper, we study the $5$-dissections of certain Ramanujan's theta functions, particularly $\psi(q)\psi(q^2), \varphi(-q)$ and $\varphi(-q)\varphi(-q^2)$, and derive an identity for $q(q;q)_{\infty}^6/(q^5;q^5)_{\infty}^6$ in terms of…
The Gordon-Andrews identities, which generalize the Rogers-Ramanujan-Schur identities, provide product and fermionic forms for the characters of the minimal conformal field theories (CFTs) M(2,2k+1). We discuss/conjecture identities of a…
In this paper, by applying a range of classic summation and transformation formulas for basic hypergeometric series, we obtain a three-term identity for partial theta functions. It extends the Andrews-Warnaar partial theta function…
In this paper we give a combinatorial proof and refinement of a Rogers-Ramanujan type partition identity of Siladi\'c arising from the study of Lie algebras. Our proof uses generating functions and $q$-difference equations.
The $\mathrm{A}_2$ Bailey chain of Andrews, Schilling and the author is extended to a four-parameter $\mathrm{A}_2$ Bailey tree. As main application of this tree, we prove the Kanade-Russell conjecture for a three-parameter family of…
In this work we study the Plancherel-Rotach type asymptotics for $q$-Laguerre orthogonal polynomials with complex scaling . The main term of the asymptotics contains Ramanujan function $A_{q}(z)$ for the scaling parameter on the vertical…
A new class of bivariate poly-analytic Hermite polynomials is considered. We show that they are realizable as the Fourier-Wigner transform of the univariate complex Hermite functions and form a nontrivial orthogonal basis of the classical…
Using results from Ramanujan's lost notebook, Zudilin recently gave an insightful proof of a radial limit result of Folsom, Ono, and Rhoades for mock theta functions. Here we see that the author's previous work on the dual nature of…
We present two general finite extensions for each of the two Rogers-Ramanujan identities. Of these one can be derived directly from Watson's transformation formula by specialization or through Bailey's method, the second similar formula can…
This is the first in a sequence of papers to develop the theory of levels in quantum K-theory and study its applications. Our main results in this paper are toric mirror theorems for permutation-equivariant quantum K-theory with level…
We study the category O of representations over a shifted Yangian. This category has a tensor product structure and contains distinguished modules, the positive prefundamental modules and the negative prefundamental modules. Motivated by…
Let R=F[D;sigma,delta] be the ring of Ore polynomials over a field (or skew field) F, where sigma is a automorphism of F and delta is a sigma-derivation. Given a an m by n matrix A over R, we show how to compute the Hermite form H of A and…
Reshetikhin-Turaev (a.k.a. Chern-Simons) TQFT is a functor that associates vector spaces to two-dimensional genus g surfaces and linear operators to automorphisms of surfaces. The purpose of this paper is to demonstrate that there exists a…
In this article we present evaluations of continued fractions studied by Ramanujan. More precisely we give the complete polynomial equations of Rogers-Ramanujan and other continued fractions, using tools from the elementary theory of the…