相关论文: Combinatorial formula for Macdonald polynomials, B…
The Hermite interpolation formulas are based on the interpretation of interpolation nodes as roots of suitable polynomials. Therefore, such formulas belong to the class of algebraic interpolations. The article considers a multidimensional…
We introduce a multivariate analogue of Bernoulli polynomials and give their fundamental properties: difference and differential relations, symmetry, explicit formula, inversion formula, multiplication theorem, and binomial type formula.…
We derive a matrix product formula for symmetric Macdonald polynomials. Our results are obtained by constructing polynomial solutions of deformed Knizhnik--Zamolodchikov equations, which arise by considering representations of the…
We give a detailed description of the nested algebraic Bethe ansatz. We consider integrable models with a $\mathfrak{gl}_3$-invariant $R$-matrix as the basic example, however, we also describe possible generalizations. We give recursions…
We define one-dimensional particles with generalized exchange statistics. The exact solution of a Hubbard-type Hamiltonian constructed with such particles is achieved using the Coordinate Bethe Ansatz. The chosen deformation of the…
We discover a family $A$ of sixteen statistics on fillings of any given Young diagram and prove new combinatorial formulas for modified Macdonald polynomials, that is, $$\tilde{H}_{\lambda}(X;q,t)=\sum_{\sigma\in…
Formulas of Rodrigues-type for the Macdonald polynomials are presented. They involve creation operators, certain properties of which are proved and other conjectured. The limiting case of the Jack polynomials is discussed.
We consider finite iterated generalized harmonic sums weighted by the binomial $\binom{2k}{k}$ in numerators and denominators. A large class of these functions emerges in the calculation of massive Feynman diagrams with local operator…
Interpolation polynomials were introduced by Knop--Sahi in type $A$, and Okounkov in type $BC$. They are inhomogeneous polynomials whose top terms are Jack and Macdonald polynomials. Thus the expansion coefficients for the product of two…
In the case of two combinatorial polynomials, we show that they can exhibited as moments of paramaterized families of orthogonal polynomials, and hence derive their Hankel transforms. Exponential Riordan arrays are the main vehicles used…
We present several new and compact formulas for the modified and integral form of the Macdonald polynomials, building on the compact "multiline queue" formula for Macdonald polynomials due to Corteel, Mandelshtam, and Williams. We also…
A filtration of a representation whose successive quotients are isomorphic to Demazure modules is called an excellent filtration. In this paper we study graded multiplicities in excellent filtrations of fusion products for the current…
The main purpose of this paper is to construct not only generating functions of the new approach Genocchi type numbers and polynomials but also interpolation function of these numbers and polynomials which are related to a, b, c arbitrary…
We present a slightly more general version of Boole's additive formula for factorials as a simple consequence of Lagrange's Interpolating Polynomial theorem.
Intermediate Macdonald polynomials for an affine root system $S$ with fixed origin and finite Weyl group $W_0$ are orthogonal polynomials invariant under a parabolic subgroup $W_J\le W_0$. The extreme cases of $W_J=1$ and $W_J=W_0$…
We consider the problem of uniform interpolation of functions with values in a complex inner product space of finite dimension. This problem can be casted within a modified weighted pluripotential theoretic framework. Indeed, in the…
In this article, we introduce combinatorial models for poly-Bernoulli polynomials and poly-Euler numbers of both kinds. As their applications, we provide combinatorial proofs of some identities involving poly-Bernoulli polynomials.
Exploiting the fact that the $q$-Whittaker polynomials arise as a specialization of the (modified) Macdonald polynomials, we derive some of their basic properties, and explore interesting identities that they satisfy. We also show how they…
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…
We give a short proof of the inner product conjecture for the symmetric Macdonald polynomials of type $A_{n-1}$. As a special case, the corresponding constant term conjecture is also proved.