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We find a simple product formula for the characteristic polynomial of the permutations with a fixed descent set under the weak order. As a corollary we obtain a simple product formula for the characteristic polynomial of alternating…
We construct double Grothendieck polynomials of classical types which are essentially equivalent to but simpler than the polynomials defined by A.N.Kirillov in arXiv:1504.01469 and identify them with the polynomials defined by T.Ikeda and…
In this paper, we study properties of polynomials over division rings. Moreover, we present formulas for finding roots of some polynomials
It is known that orthogonal polynomials obey a 3 terms recursion relation, as well as a 2x2 differential system. Here, we give an explicit and concise expression of the differential system in terms of the recursion coefficients. This result…
We derive a new explicit formula in terms of sums over graphs for the $n$-point correlation functions of general formal weighted double Hurwitz numbers coming from the Kadomtsev-Petviashvili tau functions of hypergeometric type (also known…
We obtain an effective analytic formula, with explicit constants, for the number of distinct irreducible factors of a polynomial $f \in \mathbb{Z}[x]$. We use an explicit version of Mertens' theorem for number fields to estimate a related…
We prove a closed formula for the derivative, of any order, of a implicit function, in terms of some binomial building blocks, and explain the combinatorics behind the coefficients appearing in the formula.
An observation on Hall-Littlewood polynomials.
We give an Euler-Maclaurin formula with remainder for the weighted sum of the values of a smooth function on the integral points in a simple integral polytope. Our work generalizes the formula obtained by Karshon, Sternberg and Weitsman in…
We prove a formula for the structure sheaf of a quiver variety in the Grothendieck ring of its embedding variety. This formula generalizes and gives new expressions for Grothendieck polynomials. We furthermore conjecture that the…
In the paper, the authors first inductively establish explicit formulas for derivatives of the arc sine function, then derive from these explicit formulas explicit expressions for a family of Bell polynomials related to the square function,…
We provide a formula for the Dubrovnik polynomial of a rational knot in terms of the entries of the tuple associated with a braid-form diagram of the knot. Our calculations can be easily carried out using a computer algebra system.
Simple formulas are given for generating Chern-Simons basic invariant polynomials by repeated exterior differentiation for n-dimensional differentiable manifolds having a general linear connection.
We survey several results connecting combinatorics and Wronskian solutions of the KP equation, contextualizing the successes of a recent approach introduced by Kodama, et. al. We include the necessary combinatorial and analytical background…
We obtain a recurrence relation for the f-polynomial of Gelfand-Zetlin polytopes by analyzing geometric properties of a linear projection of the Gelfand-Zetlin polytope onto a cube. We apply this recurrence relation to find explicit…
The explicit formula, which expresses the Alexander polynomials \Delta_{n,3}(t) of torus knots T(n,3) as a sum of the Alexander polynomials \Delta_{k,2}(t) of torus knots T(k,2), is found. Using this result and those from our previous…
In this paper, we discuss a proof of the isotopy invariance of a parametrized Khovanov link homology including categorifications of the Jones polynomial and the Kauffman bracket polynomial though it is a known fact. In order to present a…
We give a simple and entirely elementary proof of Gasper's theorem on the Markov sequence problem for Jacobi polynomials. It is based on the spectral analysis of an operator that arises in the study of a probabilistic model of colliding…
In this note we give the answer to the question posed by V. N. Dubinin concerning covering properties of complex polynomials
We derive a formula for the weight system of the multivariable Alexander polynomial using determinants, show that it obeys known relations, and satisfies some of the same relations as the single variable polynomial.