Related papers: Some remarks on non-symmetric interpolation Macdon…
In this paper, we develop the theory of symmetric triads with multiplicities. First, we classify abstract symmetric triads with multiplicities. Second, we determine the symmetric triads with multiplicities corresponding to commutative…
Recently we introduced a new class of relations for Bernoulli symmetric polynomials. This manuscript shows that these relations are valid for arbitrary homogeneous symmetric polynomial. Analysis of these relations leads to the discovery of…
We lift to the multivariate Eulerian polynomials the identity implying that univariate Eulerian polynomials are palindromic. As a consequence of this generalization, we obtain nice combinatorial identities that can be directly extracted…
We derive inversion formulas involving orthogonal polynomials which can be used to find coefficients of differential equations satisfied by certain generalizations of the classical orthogonal polynomials. As an example we consider special…
The usual univariate interpolation problem of finding a monic polynomial f of degree n that interpolates n given values is well understood. This paper studies a variant where f is required to be composite, say, a composition of two…
In this paper, we study some properties of associated sequaences in umbral calculus. From these properties, we derive new and interesting identities of several kinds of polynomials.
Some applications of a result, which is proved recently, is considered. We first prove three determinantal identities concerning the binomial coefficient and Stirling numbers of the first and the second kind. We also easily obtain the…
Our main aim with these notes is to introduce the combinatorial and symmetric function tools that relate to the description of the Poincare polynomial of the triply graded Khovanov-Rozansky homology of torus links, a.k.a. the (reduced)…
We provide the first (non-labelled) sequent calculi for bimodal provability logics with "usual" provability predicates. In particular, we introduce calculi for the logics CS, CSM and ER. Additionally, we present non-wellfounded versions of…
In this paper, we consider several special polynomials related to associated sequences of polynomials. Finally, we give some new and interesting identities of those polynomials arising from transfer formula for the associated sequences.
The one-variable non-symmetric Wilson polynomials are shown to coincide with the Bannai-Ito polynomials. The isomorphism between the corresponding degenerate double affine Hecke algebra of type $(C_1^{\vee}, C_1)$ and the Bannai-Ito algebra…
The incomplete version of the Macdonald function has various appellations in literature and earns a well-deserved reputation of being a computational challenge. This paper ties together the previously disjoint literature and presents the…
We establish new properties of inhomogeneous spin $q$-Whittaker polynomials, which are symmetric polynomials generalizing $t=0$ Macdonald polynomials. We show that these polynomials are defined in terms of a vertex model, whose weights come…
Based on a generalized Newton's identity, we construct a family of symmetric functions which deform the modular Hall-Littlewood functions. We also give a determinant formula for the Macdonald functions.
We give a solution to Pick's interpolation problem on the unit polydisc in $\mathbb{C}^n$, $n\geq 2$, by characterizing all interpolation data that admit a $\mathbb{D}$-valued interpolant, in terms of a family of positive-definite kernels…
A one-parameter generalisation R_{\lambda}(X;b) of the symmetric Macdonald polynomials and interpolations Macdonald polynomials is studied from the point of view of branching rules. We establish a Pieri formula, evaluation symmetry,…
We investigate the homogeneous symmetric Macdonald polynomials $P_\lambda(\X;q,t)$ for the specialization $t=q^k$. We show an identity relying the polynomials $P_\lambda(\X;q,q^k)$ and $P_\lambda(\frac{1-q}{1-q^k}\X;q,q^k)$. As a…
The aim of this short note is to show how can be derived from the properties of fundamental interpolation polynomials some nice identities.
Nonclassical symmetries and reductions of polynomial equations and systems of polynomial equations are considered. It is shown that specific polynomial equations having "hidden" symmetries can be reduced to classical symmetric systems of…
We introduce generalization of famous Macdonald polynomials for the case of super-Young diagrams that contain half-boxes on the equal footing with full boxes. These super-Macdonald polynomials are polynomials of extended set of variables:…