The number of certain integral polynomials and nonrecursive sets of integers, Part 2

We present some examples of mathematically natural nonrecursive sets of integers and relations on integers by combining results from Part 1, from recursion theory, and from the negative solution to Hilbert's 10th Problem.

     

Cardano's formula, square roots, Chebyshev polynomials and radicals

This paper is focused on the adaptation of Cardano's approach to generating the roots of rescaled Vieta-Lucas polynomials and Vieta-Fibonacci functions. The application of the derived equations to generating radical-type identities is also presented.     

The Multivariate Integer Chebyshev Problem

The multivariate integer Chebyshev problem is to find polynomials with integer coefficients that minimize the supremum norm over a compact set in ℂ ^d . We study this problem on general sets but devote special attention to product sets such as cube and polydisk. We also establish a multivariate analog of the Hilbert–Fekete upper bound for the integer Chebyshev constant, which depends on the dimension of space. In the case of single-variable polynomials in the complex plane, our estimate coincides with the Hilbert–Fekete result.      

Spectra of certain types of polynomials and tiling of integers with translates of finite sets

We investigate Fuglede's spectral set conjecture in the special case when the set in question is a union of finitely many unit intervals in dimension 1. In this case, the conjecture can be reformulated as a statement about multiplicative properties of roots of associated with the set polynomials with (0,1) coefficients. Let be an N-term polynomial. We say that {θ₁,θ₂,…,θ_N−1} is an N-spectrum for A(x) if the θ_j are all distinct and

     

关于勒让德多项式与契贝谢夫多项式间的关系

主要研究勒让德多项式与契贝谢夫多项式之间的关系的性质,利用生成函数和函数级数展开的方法,得出了勒让德多项式与契贝谢夫多项式之间的一个重要关系,这对勒让德多项式与契贝谢夫多项式的研究有一定的推动作用．     

On connection coefficients of some perturbed of arbitrary order of the Chebyshev polynomials of second kind

Orthogonal polynomials satisfy a recurrence relation of order two defined by two sequences of coefficients. If we modify one of these recurrence coefficients at a certain order, we obtain the so-called perturbed orthogonal sequence. In this work, we analyse perturbed Chebyshev polynomials of second kind and we deal with the problem of finding the connection coefficients that allow us to write the perturbed sequence in terms of the original one and in terms of the canonical basis. From the connection coefficients obtained, we derive some results about zeros at the origin. The analysis is valid for arbitrary order of perturbation.     

关于代数多项式的一个积分表示式

研究了代数多项式导数的Bernstein不等式和Markov不等式．通过代数多项式导数的一个积分表示式,给出这两个著名不等式以及它们的离散形式的证明．     

Extremal properties of certain trigonometric functions and Chebyshev polynomials

For a wide class of symmetric trigonometric polynomials, the minimal deviation property is established. As a corollary, the extremal property is proved for the components of the Chebyshev polynomial mappings corresponding to symmetric algebras A _α.     

The number of convex polyominoes and the generating function of Jacobi polynomials

Lin and Chang gave a generating function of convex polyominoes with an m+1 by n+1 minimal bounding rectangle. Gessel showed that their result implies that the number of such polyominoes is

     

Special linear combinations of orthogonal polynomials

The paper lists a number of problems that motivate consideration of special linear combinations of polynomials, orthogonal with the weight p(x) on the interval (a,b). We study properties of the polynomials, as well as the necessary and sufficient conditions for their orthogonality. The special linear combinations of Chebyshev orthogonal polynomials of four kinds with absolutely constant coefficients hold a distinguished place in the class of such linear combinations.     

On a quadrature formula of Micchelli and Rivlin

Micchelli and Rivlin (1972) obtained a quadrature formula of highest algebraic degree of precision for the Fourier-Chebyshev coefficients A_n(f), which is based on the divided differences of f′ at the zeros of the Chebyshev polynomial T_n(x). We give here a simple approach to questions of this type, which applies to the coefficients in arbitrary orthogonal expansion of f. As an auxiliary result we obtain a new interpolation formula and a new representation of the Turán quadrature formula.     

Resultants and discriminants of Chebyshev and related polynomials

We show that the resultants with respect to of certain linear forms in Chebyshev polynomials with argument are again linear forms in Chebyshev polynomials. Their coefficients and arguments are certain rational functions of the coefficients of the original forms. We apply this to establish several related results involving resultants and discriminants of polynomials, including certain self-reciprocal quadrinomials.

     

A new extension of Hermite matrix polynomials and its applications

In this paper, an extension of the Hermite matrix polynomials is introduced. Some relevant matrix functions appear in terms of the two-variable Hermite matrix polynomials. Furthermore, in order to give qualitative properties of this family of matrix polynomials, the Chebyshev matrix polynomials of the second kind are introduced.     

The Haruki-Rassias and related integral representations of the Bernoulli and Euler polynomials

Haruki and Rassias [H. Haruki, T.M. Rassias, New integral representations for Bernoulli and Euler polynomials, J. Math. Anal. Appl. 175 (1993) 81-90] found the integral representations of the classical Bernoulli and Euler polynomials and proved them by making use of the properties of certain functional equation. In this sequel, we rederive, in a completely different way, the results of Haruki and Rassias and deduce related and new integral representations. Our proofs are quite simple and remarkably elementary.     

A class of generalized complex Hermite polynomials

A class of generalized complex polynomials of Hermite type, suggested by a special magnetic Schrödinger operator, is introduced and some related basic properties are discussed.     

Cauchy, Ferrers-Jackson and Chebyshev polynomials and identities for the powers of elements of some conjugate recurrence sequences

In this paper some decompositions of Cauchy polynomials, Ferrers-Jackson polynomials and polynomials of the form x ²ⁿ + y ²ⁿ , n ∈ ℕ, are studied. These decompositions are used to generate the identities for powers of Fibonacci and Lucas numbers as well as for powers of the so called conjugate recurrence sequences. Also, some new identities for Chebyshev polynomials of the first kind are presented here.     

The zeros of Faber polynomials generated by an -star

It is shown that the zeros of the Faber polynomials generated by a regular -star are located on the -star. This proves a recent conjecture of J. Bartolomeo and M. He. The proof uses the connection between zeros of Faber polynomials and Chebyshev quadrature formulas.