Zeros of jacobisobolev orthogonal polynomials following. One of the families is orthogonal with respect to the measure d. Monotonicity of the zeros of orthogonal polynomials. Interlacing results for the zeros of di erent sequences of q orthogonal sequences with shifted parameters are given in 12,17,24,29. The behavior of their zeros, directly or indirectly, is the main reason by which popuc have received signi.
The interlacing of zeros of quasiorthogonal meixner polynomials mn x. Zeros of orthogonal polynomials aimsvolkswagen stiftung. Orthogonal polynomials and the interlacing of zeros kathy. This procedure employs gaussian rules, using interlacing properties of the zeros of orthogonal polynomials 3 and recent results about the lagrange interpolation 4. Orthogonal polynomials, linear combinations, linear functionals, recurrence relations, zeros. The first part concerns polynomials in one variable with all real roots. We also shall use a simple lemma concerning the behavior of the zeros of linear combinations of two polynomials with interlacing zeros. Zeros of geronimus perturbed orthogonal polynomials 3 using a similar approach as was done in 14, we provide a new connection formula for the geronimus perturbed mops, which will be crucial to obtain sharp limits and the speed of convergence to them of their zeros. Finally, we give necessary and su cient conditions in terms of nin order the least zero of any laguerresobolev type orthogonal polynomial be negative. We consider linear transformations that preserve real rootedness, as well as matrices that preserve interlacing. Pdf interlacing of zeros of orthogonal polynomials under. On polynomials with interlacing zeros springerlink.
These equations are used to investigate interlacing properties of zeros of sequences of q orthogonal polynomials. We give special importance to the method based on differentialdifference relation structure relation satisfied by semiclassical orthogonal polynomials. Another classical result on interlacing of zeros of orthogonal polynomials is due to stieltjes who. In the cases where zeros do not interlace, we give some numerical examples to illustrate this. Eigenvalue estimates for nonnormal matrices and the zeros of random orthogonal polynomials on the unit circle j. For related work on connections between orthogonal polynomials, their zeros, and their recurrence coef. The study of the zeros of orthogonal polynomials has a rich history s75 stimulated, in particular, by its relevance for the theory of numerical approximation g04. The zeros of orthogonal polynomials on the real line are simple, lie in the interior of the convex hull of the support of the measure and the zeros of consecutive orthogonal polynomials interlace. We show that the zeros of consecutive orthogonal polynomials pn and pn. Interlacing theo rems for the zeros of orthogonal polynomials. Interlacing of zeros of orthogonal polynomials under. In this paper, we propose a new method to approximate hvzf. Existence, real character, location and interlacing properties for the zeros of these jacobisobolev orthogonal polynomials are deduced. Muldoon department of mathematics york university north york, ont.
Interlacing theorems for the zeros of orthogonal polynomials. Using the qversion of zeilbergers algorithm, we provide a procedure to find mixed recurrence equations satisfied by classical q orthogonal polynomials with shifted parameters. Properties like orthogonality and interlacing of zeros are presented. Citeseerx document details isaac councill, lee giles, pradeep teregowda.
Jie shen department of mathematics purdue university n. Zeros of jacobi polynomials and associated inequalities. Properties of orthogonal polynomials university of kent blogs. It induces a notion of orthogonality in the usual way, namely that two polynomials are orthogonal if their inner product is zero then the sequence p n n0. Finally, we investigate the interlacing of zeros of polynomials of consecutive degree in the sequences rn. Interlacing of the zeros of jacobi polynomials with. Interlacing properties of the zeros of the orthogonal. We investigate the mutual location of the zeros of two families of orthogonal polynomials. Quasi orthogonal polynomials arise in a natural way in the context of classical orthogonal polynomials that depend on one or more parameters. Interlacing zeros of linear combinations of classical orthogonal polynomials by. We also prove that this method is numerically stable and convergent. Introduction orthogonal polynomials ops and their generalization obey and are characterized by local recurrence relations. Hahn polynomials and interlacing results for jacobi polynomials 3,7, krawtchouk polynomials 6,16 and meixner and meixnerpollaczek polynomials 16 followed.
Lemma 1 let h nx 1nax x 1 x x n and g nx 1n 1bx y 1 x y n be polynomials with real zeros, where aand bare real positive constants. Interlacing properties of zeros of multiple orthogonal. Interlacing theorems for the zeros of some orthogonal polynomials. Stieltjes interlacing of zeros of classical orthogonal.
Introduction let p pix be a family of orthogonal polynomials satisfying the three term recurrence relation. Interlacing of zeros of orthogonal polynomials under modification of the measure article pdf available in journal of approximation theory 175. Interlacing theorems for the zeros of some orthogonal. Interlacing of zeros of orthogonal polynomials under modification of. This paper examines interlacing properties of the zeros of linear. The zeros of linear combinations of orthogonal polynomials core. Pdf interlacing of zeros of shifted sequences of one. We give similar results for the zeros of pn and the associated polynomial p1 n. The integral zeros for two families of qkrawtchouk polynomials are classi. The manner in which the zeros of a polynomial change as the parameter changes can be used to study comparison and interlacing properties of the zeros 1721. The interlacing of the zeros of orthogonal polynomials of consecutive degree, pn and pn. Legendre polynomials are also useful in expanding functions of the form this is the same as before, written a little differently.
Mixed recurrences, interlacing properties and bounds of zeros. Interlacing zeros of linear combinations of classical orthogonal. The zeros of orthogonal polynomials interlace as a consequence of the. A study with \mathematica 3 in this paper we shall be concerned mainly with the method i of constructing the moments around the origin 4, showing how they can be builtup in the \mathematica symbolic package context 16. Interlacing of zeros of linear combinations of classical. We provide a comprehensive study of the zeros in terms of the free parameter of. Skew orthogonal polynomials, orthogonal polynomials, symplectic matrices, butter. In this expository paper, linear combinations of orthogonal polynomials are considered. A note on the interlacing of zeros and orthogonality.
We study the interlacing properties of the zeros of orthogonal polynomials p n and r m, m n or n. Interlacing properties of zeros for the derivative. Zero spacings of paraorthogonal polynomials on the unit. Sobolev orthogonal polynomials, jacobi orthogonal polynomials, zeros of orthogonal polynomials. The zeros of generalized krawtchouk polynomials are studied. In particular, all the roots of t n are real and lie in the interval. Christoffeldarboux formula or the recurrence relation e. Contents lists available at sciverse sciencedirect.
Interlacing zeros of linear combinations of classical. Of special interest are the zeros of the classical families of hypergeometric orthogonal polynomials, which have been fruitfully analyzed e. It means that they are eigenvectors of structured matrices. Ams proceedings of the american mathematical society. The lefthand side of the equation is the generating function for the legendre polynomials as an example, the electric potential.
Interlacing properties of zeros of multiple orthogonal polynomials. We do not discuss other methods such as those based on recurrence relations. We recall known results and some recursion relations for multiple orthogonal polynomials. M3j 1p3 canada june 1989 abstract this is a survey of some methods for. Orthogonal and biorthogonal polynomials in the theory of. Jacobi, laguerre, hermite and therefore solutions of the differential equation. We also investigate interlacing properties satisfied by the zeros of equal degree jacobi polynomials pn.
We study the interlacing properties of the zeros of orthogonal polynomials pn and rm, m n or n 1 where fpng1 n1 and frmg1 m1 are di erent sequences of orthogonal polynomials. Interlacing properties of zeros of quasi orthogonal and orthogonal jacobi polynomials o f the same or consecutive degree were discussed in 5 wher e the following result was proved. Interlacing of zeros of shifted sequences of oneparameter orthogonal polynomials article pdf available in numerische mathematik 1074. Majorizationresultsforzerosoforthogonal polynomials. The second part covers polynomials in several variables that generalize polynomials with all real roots. We discuss the extent to which the interlacing of zeros can be proved in many. The purpose of the paper is to give information about zeros of orthogonal polynomials. We use this relation to study the monotonicity properties of the zeros of generalized orthogonal polynomials. The results obtained extend a conjecture by askey, that the zeros of jacobi polynomials pn p. These topics will include the interlacing of zeros and the inequality conjectures associated with the zeros.
If, for any real constant c0, the polynomial fx h n. Monotonicity of zeros of polynomials orthogonal with. In this paper we give necessary and sufficient conditions such that the zeros of py and p, x2 strictly interlace on. This dissertation aims to study aspects of jacobi polynomials and their zeros. The classical orthogonal polynomials are often thought to be the jacobi. For related work on connections between orthogonal polynomials, their zeros, and their recurrence coefficients, see. We give an example to illustrate that the interlacing of zeros of monic polynomials of adjacent degree in a sequence is a far weaker property than the orthogonality of such a sequence. Monotonicity of zeros of laguerresobolevtype orthogonal. Relation between two sequences of orthogonal polynomials, where the associated measures are related to each other by a. Interlacing properties of zeros of associated polynomials. Mixed recurrence equations and interlacing properties for. Request pdf interlacing properties of zeros of multiple orthogonal polynomials it is well known that the zeros of orthogonal polynomials interlace.
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