Related papers: Finite Rogers--Ramanujan type identities
We generalize the theory of linked partition ideals due to Andrews using finite automata in formal language theory and apply it to prove three Rogers--Ramanujan type identities of modulo 14 that were posed by Nandi through vertex operator…
It is known that $q$-orthogonal polynomials play an important role in the field of $q$-series and special functions. During studying Dyson's "favorite" identity of Rogers--Ramanujan type, Andrews pointed out that the classical orthogonal…
The primary purpose of this paper is to provide a survey of properties, values, identities, and generalizations of the Rogers--Ramanujan continued fraction, which is closely related to the Rogers--Ramanujan identities. Many of these results…
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.
In this paper, we find an identity which connects the overpartition function and the function of Rogers--Ramanujan--Gordon type overpartitions by considering the weights and gaps. This identity can be seen as an analogue of the weighted…
There are three kinds of multiple polylogarithms; complex, finite and symmetric. The dualities for the complex and finite cases are known. In this paper, we present proofs of them via iterated integrals and its symmetric counterpart by a…
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…
AGT correspondence and its generalizations attracted a great deal of attention recently. In particular it was suggested that $U(r)$ instantons on $R^4/Z_p$ describe the conformal blocks of the coset ${\cal A}(r,p)=U(1)\times sl(p)_r\times…
We compute the graded polynomial identities of the infinite dimensional upper triangular matrix algebra over an arbitrary field. If the grading group is finite, we prove that the set of graded polynomial identities admits a finite basis. We…
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…
In 1914, Ramanujan gave a list of 17 identities expressing $1/\pi$ as linear combinations of values of hypergeometric functions at certain rational numbers. Since then, identities of similar nature have been discovered by many authors.…
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…
Ramanujan recorded five $q$-series identities at the end of his second notebook and an unified generalization of these identities obtained by Bhoria, Eyyunni and Maji. Recently, Dixit and Patel gave a finite analogue of the identity of…
Page 27 of Ramanujan's Lost Notebook contains a beautiful identity which not only gives, as a special case, a famous modular relation between the Rogers-Ramanujan functions $G(q)$ and $H(q)$ but also a relation between two fifth order mock…
A few years ago Foda, Quano, Kirillov and Warnaar proposed and proved various finite analogs of the celebrated Andrews-Gordon identities. In this paper we use these polynomial identities along with the combinatorial techniques introduced in…
We make a summary of the different types of proofs adding some new ideas. In addition we conjecture some relations which could be necessary in "modular type proofs" (not still found) of the Ramanujan-like series for 1/\pi^2.
In 2015 Choi, Kim, and Lovejoy studied a weighted partition function, $A_1(m)$, which counted subpartitions with a structure related to the Rogers--Ramanujan identities. They conjectured the existence of an infinite class of congruences for…
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…
The aim of this paper is to establish new series transforms of Bailey type and to show that these Bailey type transforms work as efficiently as the classical one and give not only new $q$-hypergeometric identities, converting double or…
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,…