Intersective polynomials and the primes

Intersective polynomials are polynomials in Z[x] having roots every modulus. For example, P₁(n)=n² and P₂(n)=n²−1 are intersective polynomials, but P₃(n)=n²+1 is not. The purpose of this note is to deduce, using results of Green and Tao (2006) [8] and Lucier (2006) [16], that for any intersective polynomial h, inside any subset of positive relative density of the primes, we can find distinct primes p₁,p₂ such that p₁−p₂=h(n) for some integer n. Such a conclusion also holds in the Chen primes (where by a Chen prime we mean a prime number p such that p+2 is the product of at most 2 primes).     

On the fourth-order difference equation for the associated Meixner polynomials

Three equivalent forms of the fourth-order difference equation obeyed by the associated Meixner polynomials (with a nonnegative real association parameter) are derived from a refinement of a recent result due to Letessier et al. (1996).     

A remark on congruences for coefficients of Faber polynomials

Faber polynomials appear when weight zero Hecke operators act on the modular j-invariant. They are polynomials in j with rational integral coefficents. Using the theory of p-adic modular forms we establish some congruences and divisibilities for these coefficients. 2000 Mathematics Subject Classification Primary—11F33 Supported by NSF grant DMS-0501225.     

Envelopes of splines in the projective plane

In this paper a family of curves—Riemannian cubics—inthe unit sphere and the real projective plane is investigated.Riemannian cubics naturally arise as solutions to variationalproblems in Riemannian spaces. It is remarkable to find thatan envelope of lines generated by a Riemannian cubic in onespace is (nearly) a Riemannian cubic in another space.     

The rate of convergence of q-Bernstein polynomials for

In the note, we obtain the estimates for the rate of convergence for a sequence of q-Bernstein polynomials {B_n,q(f)} for 0<q<1 by the modulus of continuity of f, and the estimates are sharp with respect to the order for Lipschitz continuous functions. We also get the exact orders of convergence for a family of functions , and the orders do not depend on α, unlike the classical case.     

The Minimality Properties of Chebyshev Polynomials and Their Lacunary Series

By considering four kinds of Chebyshev polynomials, an extended set of (real) results are given for Chebyshev polynomial minimality in suitably weighted Hölder norms on [–1,1], as well as (L ) minimax properties, and best L ₁ sufficiency requirements based on Chebyshev interpolation. Finally we establish best L _p , L and L ₁ approximation by partial sums of lacunary Chebyshev series of the form _i=0 a ⁱ _b ⁱ(x) where _n (x) is a Chebyshev polynomial and b is an odd integer 3. A complete set of proofs is provided.     

On Gauss-Type Quadrature Rules

Some Gauss-type Quadrature rules over [0, 1], which involve values and/or the derivative of the integrand at 0 and/or 1, are investigated. Our work is based on the orthogonal polynomials with respect to linear weight function ω(t): = 1 ? t over [0, 1]. These polynomials are also linked with a class of recently developed “identity-type functions”. Along the lines of Golub's work, the nodes and weights of the quadrature rules are computed from Jacobi-type matrices with simple rational entries. Computational procedures for the derived rules are tested on different integrands. The proposed methods have some advantage over the respective Gauss-type rules with respect to the Gauss weight function ω(t): = 1 over [0, 1].     

Exceptional Laurent biorthogonal polynomials through spectral transformations of generalized eigenvalue problems

A formulation is given for the spectral transformation of the generalized eigenvalue problem through the decomposition of the second-order differential operators. This allows us to construct some Laurent biorthogonal polynomial systems with gaps in the degree of the polynomial sequence. These correspond to an exceptional-type extension of the orthogonal polynomials, as an extension of the Laurent biorthogonal polynomials. Specifically, we construct the exceptional extension of the Hendriksen–van Rossum polynomials, which are biorthogonal analogs of the classical orthogonal polynomials. Similar to the cases of exceptional extensions of classical orthogonal polynomials, both state-deletion and state-addition occur.