Related papers: Rotundus: triangulations, Chebyshev polynomials, a…

When $\lambda$ is a partition, the specialized non-symmetric Macdonald polynomial $E_{\lambda}(x;q;0)$ is symmetric and related to a modified Hall--Littlewood polynomial. We show that whenever all parts of the integer partition $\lambda$ is…

We consider skew-products of quadratic maps over certain Misiurewicz-Thurston maps and study their statistical properties. We prove that, when the coupling function is a polynomial of odd degree, such a system admits two positive Lyapunov…

One parameter subgroups of the group of hyperbolons of volume one when exploited accurately allow one to define and investigate higher order hyperbolic-trigonometric generalization of corresponding polynomials. In parallel functions of…

We study multivariable Christoffel-Darboux kernels, which may be viewed as reproducing kernels for antisymmetric orthogonal polynomials, and also as correlation functions for products of characteristic polynomials of random Hermitian…

It is known that $q$-orthogonal polynomials play an important role in the field of $q$-series and special functions. During studying Dyson's "favorite" identity of Rogers--Ramanujan type, Andrews pointed out that the classical orthogonal…

We establish Diophantine type estimates on shifts of trigonometric polynomials on the torus $\mathbb{T}^d$, as well as that of their square roots. These estimates arise from the spectral analysis of the quasi-periodic Schr\"odinger and the…

In this paper, we mainly consider a combinatoric explanation for block Pfaffians in terms of non-intersecting paths, as a generalization of results obtained by Stembridge. As applications, we demonstrate how are generating functions of…

We extend the approach to ${\tau}$-functions as Widom constants developed by Cafasso, Gavrylenko and Lisovyy to orthogonal loop group Drinfeld-Sokolov hierarchies and isomonodromic deformations systems. The combinatorial expansion of the…

The set of triangulations of a cyclic polytope possesses two a priori different partial orders, known as the higher Stasheff-Tamari orders. The first of these orders was introduced by Kapranov and Voevodsky, while the second order was…

The first author introduced a sequence of polynomials (\cite{8}, sequence A174531) defined recursively. One of the main results of this study is proof of the integrality of its coefficients.

Consider the evolution $$ \frac{\pl m_\iy}{\pl t_n}=\Lb^n m_\iy, \frac{\pl m_\iy}{\pl s_n}=-m_\iy(\Lb^\top)^n, $$ on bi- or semi-infinite matrices $m_\iy=m_\iy(t,s)$, with skew-symmetric initial data $m_{\iy}(0,0)$. Then, $m_\iy(t,-t)$ is…

We introduce the notion of formally self-adjoint conformally covariant polydifferential operators and give some constructions of families of such operators. In one direction, we show that any homogeneous conformally variational scalar…

Ruth J.Lawrence introduced a notion of a 3-algebra to construct invariants of 3-manifolds based on their triangulations in her paper "Algebras and triangular relations". Her primary definition is suitable for certain triangulations only…

We study the recurrence coefficients of the orthogonal polynomials with respect to a semi-classical extension of the Krawtchouk weight. We derive a coupled discrete system for these coefficients and show that they satisfy the fifth…

We define the convolved h(x)-Fibonacci polynomials as an extension of the classical convolved Fibonacci numbers. Then we give some combinatorial formulas involving the h(x)-Fibonacci and h(x)-Lucas polynomials. Moreover we obtain the…

A class P_{n,m,p}(x) of polynomials is defined. The combinatorial meaning of its coefficients is given. Chebyshev polynomials are the special cases of P_{n,m,p}(x). It is first shown that P_{n,m,p}(x) may be expressed in terms of…

The Fibonacci polynomials are defined recursively as $f_{n}(x)=xf_{n-1}(x)+f_{n-2}(x)$, where $f_0(x) = 0$ and $f_1(x)= 1$. We generalize these polynomials to an arbitrary number of variables with the $r$-Fibonacci polynomial. We extend…

We investigate the triangulated structure of stable monomorphism categories (filtered chain categories) over a Frobenius category. The high degree of symmetry of linear quivers leads to a plethora of semiorthogonal decompositions into…

Computation of polynomial relative invariants is a classical tool in algebra. Relative differential invariants are central for the equivalence problem of geometric structures. We address the fundamental problem of finite generation of their…

The analogy between the nth power function and the nth Chebyshev polynomial is pursued, leading to consideration of Chebyshev radicals as analogous to ordinary radicals and Chebyshev exponents to ordinary exponents, and the cosine and…