Related papers: About sum rules for Gould-Hopper polynomials
For a family of near banded Toeplitz matrices, generalized characteristic polynomials are shown to be orthogonal polynomials of two variables, which include the Chebyshev polynomials of the second kind on the deltoid as a special case.…
It is proved that a certain symmetric sequence of nonnegative integers arising in the enumeration of magic squares of given size n by row sums or, equivalently, in the generating function of the Ehrhart polynomial of the polytope of doubly…
We introduce constrained polynomial zonotopes, a novel non-convex set representation that is closed under linear map, Minkowski sum, Cartesian product, convex hull, intersection, union, and quadratic as well as higher-order maps. We show…
This paper builds on the research initiated by Boyadzhiev, but introduces generalized harmonic numbers, \[ H_n(\alpha)= \sum_{k=1}^n \frac{\alpha^{k}}{k}, \] which enable the derivation of new identities as well as the reformulation of…
Orthogonal polynomials are of fundamental importance in many fields of mathematics and science, therefore the study of a particular family is always relevant. In this manuscript, we present a survey of some general results of the Hermite…
Extending results of Wyser, we determine formulas for the equivariant cohomology classes of closed orbits of certain families of spherical subgroups of $GL_n$ on the flag variety $GL_n/B$. Putting this together with a slight extension of…
The transformations of the sum identities for generalized harmonic and oscillatory numbers, obtained earlier in our recent report [1], enable us to derive the new identities expressed in terms of the corresponding square roots of x. At…
We introduce new polynomial invariants of a finite-dimensional semisimple and cosemisimple Hopf algebra A over a field by using the braiding structures of A. We investigate basic properties of the polynomial invariants including stability…
In this paper, we derive eight basic identities of symmetry in three variables related to generalized Bernoulli polynomials and generalized power sums. All of these are new, since there have been results only about identities of symmetry in…
We establish a form of the Gotzmann representation of the Hilbert polynomial based on rank and generating degrees of a module, which allow for a generalization of Gotzmann's Regularity Theorem. Under an additional assumption on the…
We discuss as a fundamental characteristic of orthogonal polynomials like the existence of a Lie algebra behind them, can be added to their other relevant aspects. At the basis of the complete framework for orthogonal polynomials we put…
Gaussian unitaries are specified by a second order polynomial in the bosonic operators, that is, by a quadratic polynomial and a linear term. From the Hamiltonian other equivalent representations of the Gaussian unitaries are obtained, such…
We study Hermitian random matrix models with an external source matrix which has equispaced eigenvalues, and with an external field such that the limiting mean density of eigenvalues is supported on a single interval as the dimension tends…
We consider random orthonormal polynomials $$ P_{n}(x)=\sum_{i=0}^{n}\xi_{i}p_{i}(x), $$ where $\xi_{0}$, . . . , $\xi_{n}$ are independent random variables with zero mean, unit variance and uniformly bounded $(2+\ep_0)$-moments, and…
We obtain new recurrence relations, an explicit formula, and convolution identities for higher order geometric polynomials. These relations generalize known results for geometric polynomials, and lead to congruences for higher order…
An identity involving symmetric sums of regularized multiple zeta-star values of harmonic type was proved by Hoffman. In this paper, we prove an identity of shuffle type. We use Bell polynomials appearing in the study of set partitions to…
We review the properties of six families of orthogonal polynomials that form the main bulk of the collection called the Askey--Wilson scheme of polynomials. We give connection coefficients between them as well as the so-called linearization…
We consider the problem of writing real polynomials as determinants of symmetric linear matrix polynomials. This problem of algebraic geometry, whose roots go back to the nineteenth century, has recently received new attention from the…
We obtain in closed form averages of polynomials, taken over hermitian matrices with the Gaussian measure involved in the Kontsevich integral, and prove a conjecture of Witten enabling one to express analogous averages with the full (cubic…
Certifying the positivity of trigonometric polynomials is of first importance for design problems in discrete-time signal processing. It is well known from the Riesz-Fej\'ez spectral factorization theorem that any trigonometric univariate…