Related papers: Generating functions of Chebyshev polynomials in t…
We give a uniform interpretation of the classical continuous Chebyshev's and Hahn's orthogonal polynomials of discrete variable in terms of Feigin's Lie algebra gl(N), where N is any complex number. One can similarly interpret Chebyshev's…
Understanding the underlying graph structure of a nonlinear map over a particular domain is essential in evaluating its potential for real applications. In this paper, we investigate the structure of the associated \textit{functional graph}…
Orbit functions of a simple Lie group/Lie algebra L consist of exponential functions summed up over the Weyl group of L. They are labeled by the highest weights of irreducible finite dimensional representations of L. They are of three…
We use generating functions to relate the expected values of polynomial factorization statistics over $\mathbb{F}_q$ to the cohomology of ordered configurations in $\mathbb{R}^3$ as a representation of the symmetric group. Our methods lead…
We determine the three fundamental invariants in the entries of a $3 \times 3 \times 3$ array over $\mathbb{C}$ as explicit polynomials in the 27 variables $x_{ijk}$ for $1 \le i, j, k \le 3$. By the work of Vinberg on $\theta$-groups, it…
Is is shown in this paper that there is a connection between the Riemann zeta-function $\zf$ and the classical Jacobi's polynomials, i.e. the Legendre polynomials, Chebyshev polynomials of the first and the second kind,...
Generating functions for Clebsch-Gordan coefficients of osp(1|2) are derived. These coefficients are expressed as q goes to - 1 limits of the dual q-Hahn polynomials. The generating functions are obtained using two different approaches…
Enriched versions of type A Schubert polynomials are constructed with coefficients in a polynomial ring in variables $c_1, c_2, \ldots$. Specializing these variables to $0$ recovers the double Schubert polynomials of Lascoux and…
For a Lie superalgebra with Cartan matrix over a field of positive characteristic, some information about its root system in terms of the system of simple roots corresponding to the Chevalley generators is described, under certain given…
We classify the connected $3$-dimensional differentiable Bol loops $L$ having a solvable Lie group as the group topologically generated by the left translations of $L$ using $3$-dimensional solvable Lie triple systems. Together with…
Quaternionic polynomials are generated by quaternionic variables and the quaternionic product. This paper proposes the generating ideal of quaternionic polynomials in tensor algebra, finds the Groebner base of the ideal in the case of pure…
In this paper, we study generating functions of Erd\'{e}lyi's multivariate Laguerre polynomials $L_{n_1,\cdots,n_k}^{(\alpha)}(x_1,\cdots,x_k)$ with a varying complex parameter. Our main result is a multiple generating function from which…
We study weighted Chebyshev polynomials on compact subsets of the complex plane with respect to a bounded weight function. We establish existence and uniqueness of weighted Chebyshev polynomials and derive weighted analogs of Kolmogorov's…
Kantor and Trishin described the algebra of polynomial invariants of the adjoint representation of the Lie supergalgebra $gl(m|n)$ and a related algebra $A_s$ of what they called pseudosymmetric polynomials over an algebraically closed…
We investigate Chebyshev polynomials corresponding to Jacobi weights and determine monotonicity properties of their related Widom factors. This complements work by Bernstein from 1930-31 where the asymptotical behavior of the related…
A method for constructing homogeneous Lyapunov functions of degree 1 from polynomial invariant sets is presented for linear time varying systems, homogeneous dynamic systems and the class of nonlinear systems that can be represented as…
Given a polynomial f and a finite field F one can construct a directed graph where the vertices are the values in the finite field, and emanating from each vertex is an edge joining the vertex to its image under f. When f is a Chebyshev…
In this work, the complex Lie affgebra structures on three-dimensional solvable Lie algebras are completely determined.
We characterize the generalized Chebyshev polynomials of the second kind (Chebyshev-II), and then we provide a closed form of the generalized Chebyshev-II polynomials using the Bernstein basis. These polynomials can be used to describe the…
New and old results on closed polynomials, i.e., such polynomials f in K[x_1,...,x_n] that the subalgebra K[f] is integrally closed in K[x_1,...,x_n], are collected. Using some properties of closed polynomials we prove the following…