Related papers: Some $q$-hypergeometric identities associated with…
We present alternative, q-hypergeometric proofs of some polynomial analogues of classical q-series identities recently discovered by Alladi and Berkovich, and Berkovich and Garvan.
Alladi studied partition theoretic implications of a two variable generalization of the Lebesgue identity. In this short note, we focus on a slight variation of the basic hypergeometric sum that Alladi studied. We present two new partition…
We prove a new four parameter q-hypergeometric series identity from which the three parameter key identity for the Goellnitz theorem due to Alladi, Andrews, and Gordon, follows as a special case by setting one of the parameters equal to 0.…
Given integers i,j,k,L,M, we establish a new double bounded q-series identity from which the three parameter (i,j,k) key identity of Alladi-Andrews-Gordon for Goellnitz's (big) theorem follows if L, M tend to infinity. When L = M, the…
We use $q$-binomial theorem to prove three new polynomial identities involving $q$-trinomial coefficients. We then use summation formulas for the $q$-trinomial coefficients to convert our identities into another set of three polynomial…
We propose and recursively prove polynomial identities which imply Capparelli's partition theorems. We also find perfect companions to the results of Andrews, and Alladi, Andrews and Gordon involving $q$-trinomial coefficients. We follow…
In a recent paper by the authors, a bounded version of Goellnitz's (big) partition theorem was established. Here we show among other things how this theorem leads to nontrivial new polynomial analogues of certain fundamental identities of…
In a work of 1995, Alladi, Andrews, and Gordon provided a generalization of the two Capparelli identities involving certain classes of integer partitions. Inspired by that contribution, in particular as regards the general setting and the…
Alladi and Gordon introduced the method of weighted words in 1993 to prove a refinement and generalisation of Schur's partition identity. Together with Andrews, they later used it to refine Capparelli's and G\"ollnitz' identities too. In…
If one restricts an irreducible representation $V_{\lambda}$ of $Gl_{2n}$ to the orthogonal group (respectively the symplectic group), the trivial representation appears with multiplicity one if and only if all parts of $\lambda$ are even…
We propose and prove a new polynomial identity that implies Schur's partition theorem. We give combinatorial interpretations of some of our expressions in the spirit of Kur\c{s}ung\"oz. We also present some related polynomial and $q$-series…
The Alladi-Gordon identity plays an important role for the Alladi-Gordon generalization of Schur's partition theorem. By using Joichi-Stanton's insertion algorithm, we present an overpartition interpretation for the Alladi-Gordon key…
Given complex numbers $m_1,l_1$ and nonnegative integers $m_2,l_2$, such that $m_1+m_2=l_1+l_2$, for any $a,b=0, ... ,\min(m_2,l_2)$ we define an $l_2$-dimensional Barnes type q-hypergeometric integral $I_{a,b}(z,\mu;m_1,m_2,l_1,l_2)$ and…
Given two polynomials f(x) and g(x), we extend the formula expressing the remainder in terms of the roots of these two polynomials to the case where f(x) is a Laurent polynomial. This allows us to give new expressions of a Schur function,…
Motivated by Alladi's recent multi-dimensional generalization of Sylvester's classical identity, we provide a simple combinatorial proof of an overpartition analogue, which contains extra parameters tracking the numbers of overlined parts…
We prove two identities of Hall-Littlewood polynomials, which appeared recently in a paper by two of the authors. We also conjecture, and in some cases prove, new identities which relate infinite sums of symmetric polynomials and partition…
The aim of this paper is to present a general algebraic identity. Applying this identity, we provide several formulas involving the q-binomial coefficients and the q-harmonic numbers. We also recover some known identities including an…
We provide analytic proofs for the shape invariance of the recently discovered (Odake and Sasaki, Phys. Lett. B679 (2009) 414-417) two families of infinitely many exactly solvable one-dimensional quantum mechanical potentials. These…
We will prove an identity involving refined $q$-trinomial coefficients. We then extend this identity to two infinite families of doubly bounded polynomial identities using transformation properties of the refined $q$-trinomials in an…
We first prove an identity involving symmetric polynomials. This identity leads us into exploring the geometry of Lagrangian Grassmannians. As an insight applications, we obtain a formula for the integral over the Lagrangian Grassmannian of…