相关论文: Asymmetric Rogers--Ramanujan type identities. I. T…
Recently, Rosengren utilized an integral method to prove a number of conjectural identities found by Kanade and Russell. Using this integral method, we give new proofs to some double sum identities of Rogers-Ramanujan type. These identities…
The two Rogers-Ramanujan $q$-series \[ \sum_{n=0}^{\infty}\frac{q^{n(n+\sigma)}}{(1-q)\cdots (1-q^n)}, \] where $\sigma=0,1$, play many roles in mathematics and physics. By the Rogers-Ramanujan identities, they are essentially modular…
Ramanujan listed several q-series identities in his lost notebook. The most well known q-series identities are the Rogers-Ramanujan type identities which are first discovered by Rogers and then rediscovered by Ramanujan. In this paper, we…
We prove an interesting symmetric $q$-series identity which generalizes a result due to Ramanujan. A proof that is analytic in nature is offered, and a bijective-type proof is also outlined.
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 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 deduce new q-series identities by applying inverse relations to certain identities for basic hypergeometric series. The identities obtained themselves do not belong to the hierarchy of basic hypergeometric series. We extend two of our…
We state and prove a number of unilateral and bilateral $q$-series identities and explore some of their consequences. Those include certain generalizations of the $q$-binomial sum which also generalize the $q$-Airy function introduced by…
We prove seven of the Rogers-Ramanujan type identities modulo $12$ that were conjectured by Kanade and Russell. Included among these seven are the two original modulo $12$ identities, in which the products have asymmetric congruence…
Bhoria, Eyyunni and Maji recently obtained a four-parameter $q$-series identity which gives as special cases not only all five entries of Ramanujan on pages 354 and 355 of his second notebook but also allows them to obtain an analytical…
A generalized Bailey pair, which contains several special cases considered by Bailey (\emph{Proc. London Math. Soc. (2)}, 50 (1949), 421--435), is derived and used to find a number of new Rogers-Ramanujan type identities. Consideration of…
We derive by analytic means a number of bilateral identities of the Rogers--Ramanujan type. Our results include bilateral extensions of the Rogers--Ramanujan and the G\"ollnitz-Gordon identities, and of related identities by Ramanujan,…
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 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 a recent paper, Griffin, Ono and Warnaar present a framework for Rogers-Ramanujan type identities using Hall-Littlewood polynomials to arrive at expressions of the form \[\sum_{\lambda : \lambda_1 \leq m}…
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 give a combinatorial proof of the first Rogers-Ramanujan identity by using two symmetries of a new generalization of Dyson's rank. These symmetries are established by direct bijections.
We study the behavior of the signs of the coefficients of certain infinite products involving the Rogers-Ramanujan continued fraction. For example, if $$\sum_{n=0}^{\infty}A(n)q^{n}:=…
We find a $q$-analog of the following symmetrical identity involving binomial coefficients $\binom{n}{m}$ and Eulerian numbers $A_{n,m}$, due to Chung, Graham and Knuth [{\it J. Comb.}, {\bf 1} (2010), 29--38]: {equation*} \sum_{k\geq…
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…