相关论文: Identity between Restricted Cauchy Sums for the $q…

We present several identities with a form of polynomials or rational functions that involve Pochhammer and q-Pochhammer symbols and q-binomials (i.e. Gauss polynomials). All these identities were obtained by some analytical methods based on…

The $q$-Whittaker function $W_\lambda(\mathbf{x};q)$ associated to a partition $\lambda$ is a $q$-analogue of the Schur function $s_\lambda(\mathbf{x})$, and is defined as the $t=0$ specialization of the Macdonald polynomial…

Symmetric Grothendieck polynomials are analogues of Schur polynomials in the K-theory of Grassmannians. We build dual families of symmetric Grothendieck polynomials using Schur operators. With this approach we prove skew Cauchy identity and…

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…

We introduce and study a one-parameter generalization of the q-Whittaker symmetric functions. This is a family of multivariate symmetric polynomials, whose construction may be viewed as an application of the procedure of fusion from…

The celebrated Cauchy identity expresses the product of terms $(1 - x_i y_j)^{-1}$ for $(i,j)$ indexing entries of a rectangular $m\times n$-matrix as a sum over partitions $\lambda$ of products of Schur polynomials:…

We first present some identities involving the Pochhammer symbol (rising factorial). We also recall and present some new properties of the Jacobi polynomials. We use them to expand a general hypergeometric function in an orthogonal series…

We give a proof of the generalized Cauchy identity for double Grothendieck polynomials, a combinatorial interpretation of the stable double Grothendieck polynomials in terms of triples of tableaux, and an interpolation between the stable…

For the Schur polynomials bounded and unbounded generalizations of the Cauchy identities are found.

We establish Pfaffian analogues of the Cauchy--Binet formula and the Ishikawa--Wakayama minor-summation formula. Each of these Pfaffian analogues expresses a sum of products of subpfaffians of two skew-symmetric matrices in terms of a…

We derive several identities involving Ikeda and Naruse's $K$-theoretic Schur $P$- and $Q$-functions. Our main result is a formula conjectured by Lewis and the second author which expands each $K$-theoretic Schur $Q$-function in terms of…

We prove the Cauchy type identities for the universal double Schubert polynomials, introduced recently by W. Fulton. As a corollary, the determinantal formulae for some specializations of the universal double Schubert polynomials…

This paper presents a symbolic computation method for automatically transforming $q$-hypergeometric identities to $q$-binomial identities. Through this method, many previously proven $q$-binomial identities, including $q$-Saalsch\"utz's…

We prove determinantal-Pfaffian formulae that simultaneously generalise the Pfaffian minor summation formula of Ishikawa and Wakayama and Byun's recent minor summation formula. These formulae are based on factorisation formulae for the…

The Cauchy polynomials with a $q$ parameter were recently defined, and several arithmetical properties were studied. In this paper, we establish explicit formulae for computing the Cauchy polynomials with a $q$ parameter in terms of…

We introduce and study an inhomogeneous generalization of the spin $q$-Whittaker polynomials from [Borodin,Wheeler-17]. These are symmetric polynomials, and we prove a branching rule, skew dual and non-dual Cauchy identities, and an…

We elaborate on the recent suggestion to consider averaging of Cauchy identities for the Schur functions over power sum variables. This procedure has apparent parallels with the Borel transform, only it changes the number of combinatorial…

We show the equivalence of the Pieri formula for flag manifolds and certain identities among the structure constants, giving new proofs of both the Pieri formula and of these identities. A key step is the association of a symmetric function…

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…

We study the algebraic aspects of equivariant quantum cohomology algebra of the flag manifold. We introduce and study the quantum double Schubert polynomials, which are the Lascoux-Schutzenberger type representatives of the equivariant…