Interlacing properties of zeros of multiple orthogonal polynomials

It is well known that the zeros of orthogonal polynomials interlace. In this paper we study the case of multiple orthogonal polynomials. We recall known results and some recursion relations for multiple orthogonal polynomials. Our main result gives a sufficient condition, based on the coefficients in the recurrence relations, for the interlacing of the zeros of neighboring multiple orthogonal polynomials. We give several examples illustrating our result.     

Bounds and Majorization Relations for the Zeros of a Class of Fibonacci-Type Polynomials

We apply certain matrix inequalities involving eigenvalues, the numerical radius, and the spectral radius to obtain new bounds and majorization relations for the zeros of a class of Fibonacci-type polynomials. Our results improve upon some earlier bounds for the zeros of these polynomials.     

Mixed recurrence relations and interlacing of the zeros of some q-orthogonal polynomials from different sequences

We use the generating functions of some q-orthogonal polynomials to obtain mixed recurrence relations involving polynomials with shifted parameter values. These relations are used to prove interlacing results for the zeros of Al-Salam-Chihara, continuous q-ultraspherical, q-Meixner-Pollaczek and q-Laguerre polynomials of the same or adjacent degree as one of the parameters is shifted by integer values or continuously within a certain range. Numerical examples are given to illustrate situations where the zeros do not interlace.     

One-sided zeros of subsets of the semigroup of binary relations

The one-sided zeros of the elements and of the subsets of the semigroup of all binary relations on an arbitrary nonempty set, are described. For a finite set of zeros, formulas are obtained for their number.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 600–604, May, 1990.     

Zeros of a family of hypergeometric para‐orthogonal polynomials on the unit circle

Para‐orthogonal polynomials derived from orthogonal polynomials on the unit circle are known to have all their zeros on the unit circle. In this note we study the zeros of a family of hypergeometric para‐orthogonal polynomials. As tools to study these polynomials, we obtain new results which can be considered as extensions of certain classical results associated with three term recurrence relations and differential equations satisfied by orthogonal polynomials on the real line. One of these results which might be considered as an extension of the classical Sturm comparison theorem, enables us to obtain monotonicity with respect to the parameters for the zeros of these para‐orthogonal polynomials. Finally, a monotonicity of the zeros of Meixner‐Pollaczek polynomials is proved.     

Solutions of <Emphasis Type="Italic">p</Emphasis>-adic string equations

We review several purely mathematical results concerning boundary value problems for nonlinear pseudodifferential equations for p-adic closed and open strings in the tree approximation in the case d = 1. For the solutions of these problems, we present formulas establishing the relations between the numbers of their zeros, the multiplicities of the zeros, and the numbers indicating how many times the solutions change sign.     

Polynomials orthogonal with respect to discrete convolution

The concept of “Discrete Convolution Orthogonality” is introduced and investigated. This leads to new orthogonality relations for the Charlier and Meixner polynomials. This in turn leads to bilinear representations for them. We also show that the zeros of a family of convolution orthogonal polynomials are real and simple. This proves that the zeros of the Rice polynomials are real and simple.     

Ordering of nested square roots of 2 according to the Gray code

In this paper, we discuss some relations between zeros of Lucas–Lehmer polynomials and the Gray code. We study nested square roots of 2 applying a “binary code” that associates bits 0 and 1 to “plus” and “minus” signs in the nested form. This gives the possibility to obtain an ordering for the zeros of Lucas–Lehmer polynomials, which take the form of nested square roots of 2.     

Bounds and Majorization Relations for the Zeros of Polynomials

We use matrix inequalities to prove several bounds and majorization relations for the zeros of polynomials. Our results generalize the classic bound of Montel and improve some other known bounds.     

Bounds and majorization relations for the critical points of polynomials

We apply several matrix inequalities to the derivative companion matrices of complex polynomials to establish new bounds and majorization relations for the critical points of these polynomials in terms of their zeros.     

Uniqueness Theorem of Meromorphic Mappings with Moving Targets

Some uniqueness theorems of meromorphic mappings with moving targets are given under the inclusion relations between the zeros sets of meromorphic mappings.     

Bounds for extreme zeros of some classical orthogonal polynomials

We use mixed three term recurrence relations typically satisfied by classical orthogonal polynomials from sequences corresponding to different parameters to derive upper (lower) bounds for the smallest (largest) zeros of Jacobi, Laguerre and Gegenbauer polynomials.     

Asymptotic Distributions of Zeros of Quadratic Hermite–Pade Polynomials Associated with the Exponential Function

The asymptotic distributions of zeros of the quadratic Hermite--Pad\'{e} polynomials $p_{n},q_{n},r_{n}\in{\cal P}_{n}$ associated with the exponential function are studied for $n\rightarrow\infty$. The polynomials are defined by the relation $$(*)\qquad p_{n}(z)+q_{n}(z)e^{z}+r_{n}(z)e^{2z}=O(z^{3n+2})\qquad\mbox{as} \quad z\rightarrow0,$$ and they form the basis for quadratic Hermite--Pad\'{e} approximants to $e^{z}$. In order to achieve a differentiated picture of the asymptotic behavior of the zeros, the independent variable $z$ is rescaled in such a way that all zeros of the polynomials $p_{n},q_{n},r_{n}$ have finite cluster points as $n\rightarrow\infty$. The asymptotic relations, which are proved, have a precision that is high enough to distinguish the positions of individual zeros. In addition to the zeros of the polynomials $p_{n},q_{n},r_{n}$, also the zeros of the remainder term of (*) are studied. The investigations complement asymptotic results obtained in [17].     

On Linear Dependence Relations for Integer Translates of Compactly Supported Distributions

A necessary and sufficient condition for the dimension of the space of dependence relations for (multi-) integer translates of an arbitrary compactly supported distribution in terms of zeros of its Fourier transform is given. We apply this result to obtain necessary and sufficient conditions on an integer matrix X so that the space of dependence relations for the corresponding cube spline C(·|X) is finite dimensional. We are able to describe the explicit form of all dependence relations.     

A concept of nonlinear block diagonal dominance

We introduce a concept of block diagonal dominance for nonlinear functions, and discuss the relations to strictly diagonally dominant functions and M-functions. Some sufficient conditions for the new kind of functions are given. The global convergence of block asynchronous SOR-methods for finding zeros of block diagonal dominant nonlinear functions is proved.     

ZEROS OF ENTIRE SOLUTIONS TO COMPLEX LINEAR DIFFERENCE EQUATIONS＊

In this article,we study the complex oscillation problems of entire solutions to homogeneous and nonhomogeneous linear difference equations,and obtain some relations of the exponent of convergence of z...     

Choquet order for spectra of higher Lamé operators and orthogonal polynomials

We establish a hierarchy of weighted majorization relations for the singularities of generalized Lamé equations and the zeros of their Van Vleck and Heine–Stieltjes polynomials as well as for multiparameter spectral polynomials of higher Lamé operators. These relations translate into natural dilation and subordination properties in the Choquet order for certain probability measures associated with the aforementioned polynomials. As a consequence we obtain new inequalities for the moments and logarithmic potentials of the corresponding root-counting measures and their weak-^* limits in the semi-classical and various thermodynamic asymptotic regimes. We also prove analogous results for systems of orthogonal polynomials such as Jacobi polynomials.     

On Nielsen's generalized polylogarithms and their numerical calculation

The generalized polylogarithms of Nielsen are studied, in particular their functional relations. New integral expressions are obtained, and relations for function values of particular arguments are given. An Algol procedure for calculating 10 functions of lowest order is presented. The numerical values of the Chebyshev coefficients used in this procedure are tabulated. A table of the real zeros of these functions is also given.Shortened version of CERN preprint DD/CO/69/5.Supported by Swiss National Research Fund. On leave from Consejo Nacional de Investigaciones de la Republica Argentina.     

Kloosterman sum identities and low-weight codewords in a cyclic code with two zeros

We apply relations of n-dimensional Kloosterman sums to exponential sums over finite fields to count the number of low-weight codewords in a cyclic code with two zeros. As a corollary we obtain a new proof for a result of Carlitz which relates one- and two-dimensional Kloosterman sums. In addition, we count some power sums of Kloosterman sums over certain subfields.