Related papers: On a symmetric $q$-series identity
In this paper, we present a new Rogers--Ramanujan type identity for overpartitions by extending the asymmetrical version of Schur's theorem due to Lovejoy to a broader class of infinite products. More precisely, we provide a combinatorial…
In this article, a $q$-series examined by Kluyver and Uchimura is generalized. This allows us to find generalization of the identities in the random acyclic digraph studied by Simon, Crippa, and Collenberg in 1993. As one of the corollaries…
Guillera and Zudilin proved three "divergent" Ramanujan-type supercongruences by means of the Wilf-Zeilberger algorithmic technique. In this paper, we prove $q$-analogues of two of them via the $q$-WZ method. Additionally, we give…
In this work we investigate Plancherel-Rotach type asymptotics for some $q$-series as $q\to1$. These $q$-series generalize Ramanujan function $A_{q}(z)$; Jackson's $q$-Bessel function $J_{\nu}^{(2)}$(z;q), Ismail-Masson orthogonal…
We prove, by the WZ-method, some hypergeometric identities which relate ten extended Ramanujan type series to simpler hypergeometric series. The identities we are going to prove are valid for all the values of a parameter $a$ when they are…
We propose a generalization of Bailey's lemma, useful for proving $q$-series identities. As an application, generalizations of Euler's identity, the Rogers-Ramanujan identities, and the Andrews-Gordon identities are derived. This…
In this paper, by constructing some identities, we prove some $q$-analogues of some congruences. For example, for any odd integer $n>1$, we show that \begin{gather*} \sum_{k=0}^{n-1} \frac{(q^{-1};q^2)_k}{(q;q)_k} q^k \equiv (-1)^{(n+1)/2}…
Using sequences of finite length with positive integer elements and the inversion statistic on such sequences, a collection of binomial and multinomial identities are extended to their $q$-analog form via combinatorial proofs. Using the…
We examine an identity originally stated in Ramanujan's ``lost notebook'' and first proven algebraically by Andrews and combinatorially by Kim. We give two independent combinatorial proofs and interpretations of this identity, which also…
A systematic procedure for generating certain identities involving elementary symmetric functions is proposed. These identities, as particular cases, lead to new identities for binomial and q-binomial coefficients.
In this short note, we aim to discuss some summations due to Ramanujan, their generalizations and some allied series
In this work we investigate Plancherel-Rotach type asymptotics for some $q$-series as $q\to1$. These $q$-series generalize Ramanujan function $A_{q}(z)$ ($q$-Airy function), Jackson's $q$-Bessel function $J_{\nu}^{(2)}$(z;q), Ismail-Masson…
The known WZ-proofs for Ramanujan-type series related to $1/\pi$ gave us the insight to develop a new proof strategy based on the WZ-method. Using this approach we are able to find more generalizations and discover first WZ-proofs for…
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…
Ramanujan stated an identity to the effect that if three sequences $\{a_n\}$, $\{b_n\}$ and $\{c_n\}$ are defined by $r_1(x)=:\sum_{n=0}^{\infty}a_nx^n$, $r_2(x)=:\sum_{n=0}^{\infty}b_nx^n$ and $r_3(x)=:\sum_{n=0}^{\infty}c_nx^n$ (here each…
In this paper we obtain some new transformation formula for Ramanujan summation formula and also establish some eta-function identities. we also deduce a q-Gamma function identity, n q-integral and some interesting series representation.
It is shown that (two-variable generalizations of) more than half of Slater's list of 130 Rogers-Ramanujan identities (L. J. Slater, Further identities of the Rogers-Ramanujan type, \emph{Proc. London Math Soc. (2)} \textbf{54} (1952),…
The aim of this paper is to provide a new class of series identities in the form of four general results. The results are established with the help of generalizatons of the classical Kummer's summation theorem obtained earlier by Rakha and…
We study the $q$-series $F(q) = \sum q^{m(m+1)/2}/(q)_m^2$ and offer some commentary on why Ramanujan may have thought $F(q)$ does not "close."
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…