Related papers: Chebyshev polynomials and higher order Lucas Lehme…

A two-parameter sequence of orthogonal polynomials $\{P_n( x; \lambda, t)\}_{n\ge 0}$ with respect to the weight function $x^\alpha e^{- \lambda x} \rho_\nu(x t),\ \alpha > -1,\ \lambda, t \ge 0, \ \rho_{\nu}(x)= 2 x^{\nu/2} K_\nu(2\sqrt…

After fixing a triangulation $L$ of a $k$-dimensional simplex that has no new vertices on the boundary, we introduce a triangulation operation on all simplicial complexes that replaces every $k$-face with a copy of $L$, via a sequence of…

We construct admissible polynomial meshes on piecewise polynomial or trigonometric curves of the complex plane, by mapping univariate Chebyshev points. Such meshes can be used for polynomial least-squares, for the extraction of Fekete-like…

We establish the existence of infinitely many \emph{polynomial} progressions in the primes; more precisely, given any integer-valued polynomials $P_1, >..., P_k \in \Z[\m]$ in one unknown $\m$ with $P_1(0) = ... = P_k(0) = 0$ and any $\eps…

We investigate the problem of computing tensor product multiplicities for complex semisimple Lie algebras. Even though computing these numbers is #P-hard in general, we show that if the rank of the Lie algebra is assumed fixed, then there…

To study a Dirichlet polynomial $f(s)=\frac{a_{m}}{m^{s}}+\cdots +\frac{a_{n}}{n^{s}}$ by regarding it as a multivariate polynomial via the canonical map $\phi$ sending $p_i^{-s}$ to an indeterminate $X_i$, with $p_i$ the $i$th prime…

We consider a new multivariate generalization of the classical monic (univariate) Chebyshev polynomial that minimizes the uniform norm on the interval $[-1,1]$. Let $\Pi^*_n$ be the subset of polynomials of degree at most $n$ in $d$…

Let $(L_n^{(k)})_{n\geq 2-k}$ be the sequence of $k$-generalized Lucas numbers for some fixed integer $k\ge 2$, whose first $k$ terms are $0,\;\ldots\;,\;0,\;2,\;1$ and each term afterward is the sum of the preceding $k$ terms. In this…

Two sums over the inverse of the product of an integer n and its greatest prime factor G(n), are computed to first 13 decimal digits. These sums converge, but converge very slowly. They are transformed into sums involving Mertens' prime…

There is a two-component log-gas system with Boltzmann factor which provides an interpolation between the eigenvalue PDF for $\beta = 1$ and $\beta = 4$ invariant random matrix ensembles. The solvability of this log-gas system relies on the…

In this paper we generalize and specialize generating functions for classical orthogonal polynomials, namely Jacobi, Gegenbauer, Chebyshev and Legendre polynomials. We derive a generalization of the generating function for Gegenbauer…

Let $R=\mathcal{O}_{\Q(\sqrt{d})}$ for $d<0$, squarefree, $d\neq -1,-3$. We prove Lehmer's conjecture for associated reciprocal polynomials of $R$-matrices; that is, any noncyclotomic $R$-matrix has Mahler measure at least…

The direct spreading measures of the Laguerre polynomials, which quantify the distribution of its Rakhmanov probability density along the positive real line in various complementary and qualitatively different ways, are investigated. These…

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.…

In this paper we discuss some relations between zeros of Lucas-Lehmer polynomials and Gray code. We study nested square roots of 2 applying a "binary code" that associates bits $0$ and $1$ to $\oplus$ and $\ominus$ signs in the nested form.…

The Householder's method is a root-find algorithm which is a natural extension of the methods of Newton and Halley. The current paper mostly focuses on approximating the square root of a positive real number based on these methods. The…

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 $\phi$…

A composite number $n$ is called a Lehmer number when $\phi(n) | n - 1$, where $\phi$ is the Euler totient function. Lehmer's totient problem asks if there exist any composite numbers $n$ such that $\phi(n)| n-1$? No such numbers are known.…

Iteration of randomly chosen quadratic maps defines a Markov process: X_{n+1}=\epsilon_{n+1}X_n(1-X_n), where \epsilon_n are i.i.d. with values in the parameter space [0,4] of quadratic maps F_{\theta}(x)=\theta x(1-x). Its study is of…

The discrete Chebyshev polynomials $t_n(x,N)$ are orthogonal with respect to a distribution function, which is a step function with jumps one unit at the points $x=0,1,..., N-1$, N being a fixed positive integer. By using a double integral…