New inequalities from classical Sturm theorems

Inequalities satisfied by the zeros of the solutions of second-order hypergeometric equations are derived through a systematic use of Liouville transformations together with the application of classical Sturm theorems. This systematic study allows us to improve previously known inequalities and to extend their range of validity as well as to discover inequalities which appear to be new. Among other properties obtained, Szegő's bounds on the zeros of Jacobi polynomials for , are completed with results for the rest of parameter values, Grosjean's inequality (J. Approx. Theory 50 (1987) 84) on the zeros of Legendre polynomials is shown to be valid for Jacobi polynomials with |β|1, bounds on ratios of consecutive zeros of Gauss and confluent hypergeometric functions are derived as well as an inequality involving the geometric mean of zeros of Bessel functions.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">m</Emphasis></Subscript> Extemal Polynomials Associated with Generalized Jacobi Weights

Asymptotic estimations of the Christoffel type functions for Lm extremal polynomials with an even integer m associated with generalized Jacobi weights are established. Also, asymptotic behavior of the zeros of the Lm extremal polynomials and the Cotes numbers of the corresponding Turan quadrature formula is given.     

Characterizations of the Optimal Descartes' Rules of Signs

A system of functions satisfies Descartes' rule of signs if the number of zeros (with multiplicities) of a linear combination of these functions is less than or equal to the number of variations of strict sign in the sequence of the coefficients. In this paper we characterize the systems of functions satisfying a stronger property than the above mentioned Descartes' rule: The difference between the number of zeros and the changes of sign in the sequence of coefficients must be always a nonnegative even number. We show that the approximation to the number of zeros given by these systems of functions is better than the approximation provided by any other systems of functions satisfying a Descartes' rule of signs. This last result improves, in the particular case of polynomials, the main theorem of [14].     

亚纯函数与其微分多项式的分担函数与正规族

设${\cal F}$为开平面内的区域$D$上的亚纯函数族, ${\cal F}$中任何函数$f(z)\in{\cal F}$, $f$的零点竽数至少为$k+1$.对于$D$内不等于零的解析函数$a(z)$.若$f(z)$与其微分多项式$D(f)$ IM分担$a(z)$,本文不仅得到${\cal F}$在$D$上正规, 而且得到相应于正规函数的结果.     

Spectral properties of solutions of hypergeometric-type differential equations

The second-order differential equation σ(x)y″ + τ(x)y′ + λy = 0 is usually called equation of hypergeometric type, provided that σ, τ are polynomials of degree not higher than two and one, respectively, and λ is a constant. Their solutions are commonly known as hypergeometric-type functions (HTFs). In this work, a study of the spectrum of zeros of those HTFs for which , v , and σ, τ are independent of ν, is done within the so-called semiclassical (or WKB) approximation. Specifically, the semiclassical or WKB density of zeros of the HTFs is obtained analytically in a closed way in terms of the coefficients of the differential equation that they satisfy. Applications to the Gaussian and confluent hypergeometric functions as well as to Hermite functions are shown.     

A logician's view of graph polynomials

Graph polynomials are graph parameters invariant under graph isomorphisms which take values in a polynomial ring with a fixed finite number of indeterminates. We study graph polynomials from a model theoretic point of view. In this paper we distinguish between the graph theoretic (semantic) and the algebraic (syntactic) meaning of graph polynomials. Graph polynomials appear in the literature either as generating functions, as generalized chromatic polynomials, or as polynomials derived via determinants of adjacency or Laplacian matrices. We show that these forms are mutually incomparable, and propose a unified framework based on definability in Second Order Logic. We show that this comprises virtually all examples of graph polynomials with a fixed finite set of indeterminates. Finally we show that the location of zeros and stability of graph polynomials is not a semantic property. The paper emphasizes a model theoretic view. It gives a unified exposition of classical results in algebraic combinatorics together with new and some of our previously obtained results scattered in the graph theoretic literature.     

The distribution of zeros of asymptotically extremal polynomials

In this paper, we study the asymptotic behavior of the zeros of a sequence of polynomials whose weighted norms have the same nth root behavior as the weighted norms for certain extremal polynomials. Our results include as special cases several of the previous results of Erd s, Freud, Jentzsch, Szeg and Blatt, Saff, and Simkani. Applications are given concerning the zeros of orthogonal polynomials over a smooth Jordan curve (in particular, on the unit circle) and the zeros of polynomials of best approximation on R to nonentire functions.     

Orthogonal Rational Functions and Frequency Analysis

One way of finding unknown frequencies in a trigonometric signal is to use Szeg theory, where under certain conditions asymptotic behavior of zeros of Szeg polynomials lead to the frequencies. Recently this was extended to generalized Szeg theory, i.e. where polynomials are replaced by certain rational functions.This note presents a brief overview of some of the Szeg theory, including also a general formula for the monic orthogonal rational functions. Moreover, for a certain measure, constructed from the observations of the signal, the moments are explicitely determined.Finally a simple example is included, indicating the connection between location of an interpolation point and the way zeros approach frequency points.     

一类整函数系数微分方程解的复振荡

本文研究了微分方程f~(k) A_((k-1))f~((k-1)) … A_0f=F(k≥2)解的增长级和零点收敛指数,其中A_j=B_je~(P_j),j=0,1,…,k-1,B_j(z)为整函数,P_j(z)为多项式,σ(B_j)＜degP_j．     

Asymptotic analysis of the Askey-scheme I: from Krawtchouk to Charlier

We analyze the Charlier polynomials C _n (χ) and their zeros asymptotically as n → ∞. We obtain asymptotic approximations, using the limit relation between the Krawtchouk and Charlier polynomials, involving some special functions. We give numerical examples showing the accuracy of our formulas.      

Strong asymptotics for Krawtchouk polynomials

We determine the strong asymptotics for the class of Krawtchouk polynomials on the real line. We show how our strong asymptotics describes the limiting distribution of the zeros of the Krawtchouk polynomials.     

Sums of entire functions having only real zeros

We show that certain sums of products of Hermite-Biehler entire functions have only real zeros, extending results of Cardon. As applications of this theorem, we construct sums of exponential functions having only real zeros, we construct polynomials having zeros only on the unit circle, and we obtain the three-term recurrence relation for an arbitrary family of real orthogonal polynomials. We discuss a similarity of this result with the Lee-Yang Circle Theorem from statistical mechanics. Also, we state several open problems.

     

Asymptotics of orthogonal polynomials and the numerical solution of ill-posed problems

The paper reviews the impact of modern orthogonal polynomial theory on the analysis of numerical algorithms for ill-posed problems. Of major importance are uniform bounds for orthogonal polynomials on the support of the weight function, the growth of the extremal zeros, and asymptotics of the Christoffel functions.     

Erdős-Turʼan-Type Discrepancy for the Zeros of Rational Functions

In 1950 P. Erdős and P. Turán published a discrepancy theorem for the zeros of a polynomial. Therein, the maximum deviation of the normalized zero counting measure from the equilibrium measure of the unit circle is estimated. Many other discrepancy theorems and related propositions about weak-star-convergence of the zero distribution of a sequence of polynomials were proved during the last decades. For several years the weak-star-convergence of the zero distribution of a sequence of rational functions is also studied. The main result of this paper is a discrepancy theorem for the zero distribution of a rational function which generalizes and sharpens previous propositions about weak-star-convergence of the zero counting measure of sequences of rational functions and known discrepancy theorems for polynomials.     

Properties of the zeros of confluent hypergeometric functions

Several infinite systems of nonlinear algebraic equations satisfied by the zeros of confluent hypergeometric functions are derived. Certain sum rules and other related properties for the zeros follow from these equations. A large class of special functions, which are special cases of confluent hypergeometric functions, is included. This is illustrated in the case of the zeros of Bessel functions and Laguerre polynomials.     

实多项式有纯虚零点的代数判据

多项式零点分布的研究,在数学的许多分枝及工程应用中都有重要意义.其中人们最关心的问题之一是,判定一个多项式是否为 Hurwitz 多项式,即它的零点是否均具有负实部.对此著名的 Routh-Hurwitz (代数)判据和 Nyquist (频域)判据已给出了完全的解答.近年来,鲁棒反馈镇定等问题的研究,又提出了判定系数在某一范围内变动的一簇多项式的稳定性问题.这方面最引人注目的是 Kharitonov 的结果及随后人们所作的各     

Some results on zeros and uniqueness of difference-differential polynomials

We consider the zeros distributions of difference-differential polynomials which are the derivatives of difference products of entire functions.We also investigate the uniqueness problems of difference...     

Convexity of the zeros of some orthogonal polynomials and related functions

We study convexity properties of the zeros of some special functions that follow from the convexity theorem of Sturm. We prove results on the intervals of convexity for the zeros of Laguerre, Jacobi and ultraspherical polynomials, as well as functions related to them, using transformations under which the zeros remain unchanged. We give upper as well as lower bounds for the distance between consecutive zeros in several cases.     

亚纯函数微分多项式f~kQ[f]+P[f]的零点分布

文[4]对简单形式的微分多项式fkf’+a的零点分布进行了讨论,文[1]对一般形式的微分多项式fkQ[f]+P[f]的零点分布进行了讨论.但由于极点给证明带来的困难,这些工作主要是对整函数来做的.本文证明了任一满足δ(∞,f)>k+2ΓQ+3ΓP+2/2k+2ΓQ+1的超越亚纯函数f,微分多项式fkQ[f]+P[f]在不含f,Q[f]极点和P[f]零、极点的可数个圆盘并集之外有无穷多个零点,其中k≥3Γp+2,而ΓQ,ΓP分别是f的微分多项式Q[f],P[f]的权.文[1]和[2,4,6]中的结论是本文结论的特殊情况.     

Polynomials whose coefficients coincide with their zeros

In this paper we consider monic polynomials such that their coefficients coincide with their zeros. These polynomials were first introduced by S. Ulam. We combine methods of algebraic geometry and dynamical systems to prove several results. We obtain estimates on the number of Ulam polynomials of degree N. We provide additional methods to obtain algebraic identities satisfied by the zeros of Ulam polynomials, beyond the straightforward comparison of their zeros and coefficients. To address the question about the existence of orthogonal Ulam polynomial sequences, we show that the only Ulam polynomial eigenfunctions of hypergeometric type differential operators are the trivial Ulam polynomials \(\{x^N\}_{N=0}^\infty \). We propose a family of solvable N-body problems such that their stable equilibria are the zeros of certain Ulam polynomials.