Related papers: Rogers-Ramanujan identities in Statistical Mechani…

We revisit the solvable lattice models described by Andrews Baxter and Forrester and their generalizations. The expressions for the local state probabilities were shown to be related to characters of the minimal models. We recompute these…

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…

Basil Gordon, in the sixties, and George Andrews, in the seventies, generalized the Rogers-Ramanujan identities to higher moduli. These identities arise in many areas of mathematics and mathematical physics. One of these areas is…

Integer partitions have long been of interest to number theorists, perhaps most notably Ramanujan, and are related to many areas of mathematics including combinatorics, modular forms, representation theory, analysis, and mathematical…

We give several expansion and identities involving the Ramanujan function $A_q$ and the Stieltjes--Wigert polynomials. Special values of our idenitities give $m$-versions of some of the items on the Slater list of Rogers-Ramanujan type…

The Rogers-Ramanujan-Gordon identities generalize the classical partition identities discovered independently by L. J. Rogers and S. Ramanujan. In 2021, Afsharijoo provided a commutative algebra proof of the Rogers-Ramanujan-Gordon…

Many classical $q$-series identities, such as the Rogers--Ramanujan identities, yield combinatorial interpretations in terms of integer partitions. Here we consider algebraically manipulating some of the classical $q$-series to yield…

Resorting to the recursions satisfied by the polynomials which converge to the right hand sides of the Rogers-Ramanujan type identities given by Sills and a determinant method presented in a paper by Ismail-Prodinger-Stanton, we obtain many…

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…

Here we consider the $q$-series coming from the Hall-Littlewood polynomials, \begin{equation*} R_\nu(a,b;q)=\sum_{\substack{\lambda \\[1pt] \lambda_1\leq a}} q^{c|\lambda|} P_{2\lambda}\big(1,q,q^2,\dots;q^{2b+d}\big). \end{equation*} These…

We generalize the "motivated proof" of the Rogers-Ramanujan identities given by Andrews and Baxter to provide an analogous "motivated proof" of Gordon's generalization of the Rogers-Ramanujan identities. Our main purpose is to provide…

George Andrews [\emph{Bull. Amer. Math. Soc.}, 2007, 561--573] introduced the idea of a \emph{signed partiton} of an integer; similar to an ordinary integer partitions, but where some of the parts could be negative. Further, Andrews…

We use an integral method to establish a number of Rogers-Ramanujan type identities involving double and triple sums. The key step for proving such identities is to find some infinite products whose integrals over suitable contours are…

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…

Presented are polynomial identities which imply generalizations of Euler and Rogers--Ramanujan identities. Both sides of the identities can be interpreted as generating functions of certain restricted partitions. We prove the identities by…

We present proofs of two new families of sum-product identities arising from the cylindric partitions paradigm. Most of the presented expressions, the related sum-product identities, and the ingredients for the proofs were first conjectured…

We prove a number of new Rogers-Ramanujan type identities involving double, triple and quadruple sums. They were discovered after an extensive search using Maple. The main idea of proofs is to reduce them to some known identities in the…

In this note we show how to rederive the $A_2$ Rogers-Ramanujan identities proven by Andrews, Schilling and Warnaar using cylindric partitions. This paper is dedicated to George Andrews for his $80^{th}$ birthday.

We show that, in many cases, there are infinitely many sets of partitions corresponding to a single analytical Rogers-Ramanujan type identity. This means that a single analytical Rogers-Ramanujan type identity implies the existence of…

Using Lie theory, Stefano Capparelli conjectured an interesting Rogers-Ramanujan type partition identity in his 1988 Rutgers Ph.D. thesis. The first proof was given by George Andrews, using combinatorial methods. Later, Capparelli was able…