384 research outputs found
Some recent results on the zeros of Bessel functions and orthogonal polynomials
AbstractSeveral recent results are encountered on the zeros of Bessel functions and of the classical orthogonal polynomials, many of them are waiting for being published. Four problems are formulated as conjectures
Zeros of Jacobi polynomials and associated inequalities
A Dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the Degree of Master of Science. Johannesburg 2015.This Dissertation focuses on the Jacobi polynomial. Specifically, it discusses certain
aspects of the zeros of the Jacobi polynomial such as the interlacing property and quasiorthogonality.
Also found in the Dissertation is a chapter on the inequalities of the zeros
of the Jacobi polynomial, mainly those developed by Walter Gautschi
Zeros of Quasi-Orthogonal Jacobi Polynomials
We consider interlacing properties satisfied by the zeros of Jacobi
polynomials in quasi-orthogonal sequences characterised by ,
. We give necessary and sufficient conditions under which a
conjecture by Askey, that the zeros of Jacobi polynomials and are interlacing, holds when the
parameters and are in the range and .
We prove that the zeros of and
do not interlace for any ,
and any fixed , with , . The
interlacing of zeros of and for
is discussed for and in this range, , and new upper and lower bounds are derived for the zero of
that is less than
On Asymptotics of Polynomial Eigenfunctions for Exactly-Solvable Differential Operators
In this paper we study the asymptotic zero distribution of eigenpolynomials
for degenerate exactly-solvable operators. We present an explicit conjecture
and partial results on the growth of the largest modulus of the roots of the
unique and monic n:th degree eigenpolynomial of any such operator as the degree
n tends to infinity. Based on this conjecture we deduce the algebraic equation
satified by the Cauchy transform of the asymptotic root measure of the properly
scaled eigenpolynomials, for which the union of all roots is conjecturally
contained in a compact set.Comment: 36 pages, 37 figures, to appear in Journal of Approximation Theor
Inequalities for Extreme Zeros of Some Classical Orthogonal and q-orthogonal Polynomials
Let {pn}1 n=0 be a sequence of orthogonal polynomials. We briefly review properties of pn that have been used to derive upper and lower bounds for the largest and smallest zero of pn. Bounds for the extreme zeros of Laguerre, Jacobi and Gegenbauer polynomials that have been obtained using different approaches are numerically compared and new bounds for extreme zeros of q-Laguerre and little q-Jacobi polynomials are proved
Zeros of polynomials orthogonal with respect to a signed weight
AbstractIn this paper we consider the monic polynomial sequence (Pnα,q(x)) that is orthogonal on [−1,1] with respect to the weight function x2q+1(1−x2)α(1−x),α>−1,q∈N; we obtain the coefficients of the tree-term recurrence relation(TTRR) by using a different method from the one derived in Atia et al. (2002) [2]; we prove that the interlacing property does not hold properly for (Pnα,q(x)); and we also prove that, if xn,nα+i,q+j is the largest zero of Pnα+i,q+j(x), x2n−2j,2n−2jα+j,q+j<x2n−2i,2n−2iα+i,q+i,0≤i<j≤n−1
Functions preserving nonnegativity of matrices
The main goal of this work is to determine which entire functions preserve
nonnegativity of matrices of a fixed order -- i.e., to characterize entire
functions with the property that is entrywise nonnegative for every
entrywise nonnegative matrix of size . Towards this goal, we
present a complete characterization of functions preserving nonnegativity of
(block) upper-triangular matrices and those preserving nonnegativity of
circulant matrices. We also derive necessary conditions and sufficient
conditions for entire functions that preserve nonnegativity of symmetric
matrices. We also show that some of these latter conditions characterize the
even or odd functions that preserve nonnegativity of symmetric matrices.Comment: 20 pages; expanded and corrected to reflect referees' remarks; to
appear in SIAM J. Matrix Anal. App
Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions
I show that there exist universal constants such that, for
all loopless graphs of maximum degree , the zeros (real or complex)
of the chromatic polynomial lie in the disc . Furthermore,
. This result is a corollary of a more general result
on the zeros of the Potts-model partition function in the
complex antiferromagnetic regime . The proof is based on a
transformation of the Whitney-Tutte-Fortuin-Kasteleyn representation of to a polymer gas, followed by verification of the
Dobrushin-Koteck\'y-Preiss condition for nonvanishing of a polymer-model
partition function. I also show that, for all loopless graphs of
second-largest degree , the zeros of lie in the disc . Along the way, I give a simple proof of a generalized (multivariate)
Brown-Colbourn conjecture on the zeros of the reliability polynomial for the
special case of series-parallel graphs.Comment: 47 pages (LaTeX). Revised version contains slightly simplified proofs
of Propositions 4.2 and 4.5. Version 3 fixes a silly error in my proof of
Proposition 4.1, and adds related discussion. To appear in Combinatorics,
Probability & Computin
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