Related papers: On Loewner's characterization of polynomials
We consider the Ehrhart polynomial of hypersimplices. It is proved that these polynomials have positive coefficients and we give a combinatorial formula for each of them. This settles a problem posed by Stanley and also proves that uniform…
This paper proves that the characteristic polynomial is a complete unitary invariant for pairs of projection matrices. Some special cases involving three or more projections are also considered.
An overview of the basic results on Macdonald(-Koornwinder) polynomials and double affine Hecke algebras is given. We develop the theory in such a way that it naturally encompasses all known cases. Among the basic properties of the…
A new family of polynomials, called cumulant polynomial sequence, and its extensions to the multivariate case is introduced relied on a purely symbolic combinatorial method. The coefficients of these polynomials are cumulants, but depending…
The purpose of this note is to give an affirmative answer to a conjecture appearing in [Integral Transforms Spec. Funct. 26 (2015) 90-95].
We consider polynomials in R[x] which map the set of nonnegative (element-wise) matrices of a given order into itself. Let n be a positive integer and define P(n)= {p in R[x] : p(A) is nonnegative (element-wise), for all A, A an n-by-n…
In this paper, we first give a simple combinatorial proof of Tepper's identity. Then, as a by product of this interesting identity we present another proof of the well-known Wilson's identity in number theory. Finally, we obtain a…
For the solvable polynomial algebras introduced and studied by Kandri-Rody and Weispfenning [J. Symbolic Comput., 9(1990)], a constructive characterization is given in terms of Gr\"obner bases for ideals of free algebras, thereby solvable…
In 1971 Griffiths used a generating function to define polynomials in d variables orthogonal with respect to the multinomial distribution. The polynomials possess a duality between the discrete variables and the degree indices. In 2004…
We identify the Atkin polynomials in terms of associated Jacobi polynomials. Our identificationthen takes advantage of the theory of orthogonal polynomials and their asymptotics to establish many new properties of the Atkin polynomials.…
We present an elementary method for proving enumeration formulas which are polynomials in certain parameters if others are fixed and factorize into distinct linear factors over Z. Roughly speaking the idea is to prove such formulas by…
We formulate several polynomial identities. One side of these identities has a nice simple form. Whereas the other has a form of a polynomial whose coefficients contain binomial coefficients double factorials or (and) rising factorials. The…
We obtain all spectral type differential equations satisfied by the Sobolev-type Laguerre polynomials. This generalizes the results found in 1990 by the first and second author in the case of the generalized Laguerre polynomials defined by…
The M-convexity of dual Schubert polynomials was first proven by Huh, Matherne, M\'esz\'aros, and St. Dizier in 2022. We give a full characterization of the supports of dual Schubert polynomials, which yields an elementary alternative proof…
This report formulates a conjectural combinatorial rule that positively expands Grothendieck polynomials into Lascoux polynomials. It generalizes one such formula expanding Schubert polynomials into key polynomials, and refines another one…
For polynomials of degree two over finite fields, we present an improvement of Fitzgerald's characterization (Finite Fields Appl. 9(1):117-121, 2003). We then use this new characterization to obtain an explicit, complete, and simple…
Let $f \in \mathbb{R}[x]$ be a polynomial with real coefficients. We say that $f$ is eventually non-negative if $f^m$ has non-negative coefficients for all sufficiently large $m \in \mathbb{N}$. In this short note, we give a classification…
We study a class of bivariate deformed Hermite polynomials and some of their properties using classical analytic techniques and the Wigner map. We also prove the positivity of certain determinants formed by the deformed polynomials. Along…
We provide a short proof of the theorem that every real multivariate polynomial has a symmetric determinantal representation, which was first proved in J. W. Helton, S. A. McCullough, and V. Vinnikov, Noncommutative convexity arises from…
We obtain new partial results supporting the spectral set conjecture in dimension 1.