Related papers: Partial Chebyshev Polynomials and Fan Graphs
We derive a general formula for the quadratic embedding constant of a graph join $\bar{K}_m+G$, where $\bar{K}_m$ is the empty graph on $m\ge1$ vertices and $G$ is an arbitrary graph. Applying our formula to a fan graph $K_1+P_n$, where…
We solve the problem of factoring polynomials $V_n(x) \pm 1$ and $W_n(x) \pm 1$ where $V_n(x)$ and $W_n(x)$ are Chebyshev polynomials of the third and fourth kinds. The method of proof is based on previous work by Wolfram [12] for factoring…
The quadratic embedding constant (QEC) of a finite, simple, connected graph originated from the classical work of Schoenberg [Ann. of Math., 1935] and [Trans. Amer. Math. Soc., 1938] on Euclidean distance geometry. In this article, we study…
Working over a field $\kk$ of characteristic zero, this paper studies line embeddings of the form $\phi = (T_i,T_j,T_k):\A^1\to\A^3$, where $T_n$ denotes the degree $n$ Chebyshev polynomial of the first kind. In {\it Section 4}, it is shown…
For an analytic function $\phi(z)$ with a Laurent expansion at $\infty$ of the form \begin{equation*} \phi(z)=z+c_{0}+\frac{c_{1}}{z}+\frac{c_{2}}{z^{2}}+\cdots, \end{equation*} the Faber polynomial $F_n$ of degree $n$ associated to $\phi$…
Analytic expressions for the Fourier transforms of the Chebyshev and Legendre polynomials are derived, and the latter is used to find a new representation for the half-order Bessel functions. The numerical implementation of the so-called…
The quadratic embedding constant (QE constant) of a graph is a new characteristic value of a graph defined through the distance matrix. We derive formulae for the QE constants of the join of two regular graphs, double graphs and certain…
We study the problem of minimizing the supremum norm, on a segment of the real line or on a compact set in the plane, by polynomials with integer coefficients. The extremal polynomials are naturally called integer Chebyshev polynomials.…
A connected graph $G$ is of QE class if it admits a quadratic embedding in a Hilbert space, or equivalently if the distance matrix is conditionally negative definite, or equivalently if the quadratic embedding constant $\mathrm{QEC}(G)$ is…
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…
The quadratic embedding constant (QEC) of a graph $G$ is a new numeric invariant, which is defined in terms of the distance matrix and is denoted by $\mathrm{QEC}(G)$. By observing graph structure of the maximal cliques (clique graph), we…
We show that the Kronecker coefficients (the Clebsch-Gordan coefficients of the symmetric group) indexed by two two-row shapes are given by quadratic quasipolynomial formulas whose domains are the maximal cells of a fan. Simple calculations…
We obtain an explicit formula for the quadratic embedding constant (QEC) of a strongly regular graph $\mathrm{srg}(n,k,\lambda,\mu)$ with $\mu\ge1$. By using QEC we give a necessary and sufficient condition for a strongly regular graph to…
Quantum generative modeling is emerging as a powerful tool for advancing data analysis in high-energy physics, where complex multivariate distributions are common. However, efficiently learning and sampling these distributions remains…
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…
A numerical semigroup $S$ is an additively-closed set of non-negative integers, and a factorization of an element $n$ of $S$ is an expression of $n$ as a sum of generators of $S$. It is known that for a given numerical semigroup $S$, the…
We completely describe the functional graph associated to iterations of Chebyshev polynomials over finite fields. Then, we use our structural results to obtain estimates for the average rho length, average number of connected components and…
In this work, considering a general subclass of bi-univalent functions and using the Chebyshev polynomials, we obtain coefficient expansions for functions in this class.
We introduce quantized Chebyshev polynomials as deformations of generalized Chebyshev polynomials previously introduced by the author in the context of acyclic coefficient-free cluster algebras. We prove that these quantized polynomials…
Using Chebyshev polynomialsof both kinds, we construct rational fractions which are convergents of the smallest root of $x^2-\alpha x+1$ for $\alpha=3,4,5,\dots$.Some of the underlying identities suggest an identity involving…