389 research outputs found
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
A unified approach to polynomial sequences with only real zeros
We give new sufficient conditions for a sequence of polynomials to have only
real zeros based on the method of interlacing zeros. As applications we derive
several well-known facts, including the reality of zeros of orthogonal
polynomials, matching polynomials, Narayana polynomials and Eulerian
polynomials. We also settle certain conjectures of Stahl on genus polynomials
by proving them for certain classes of graphs, while showing that they are
false in general.Comment: 19 pages, Advances in Applied Mathematics, in pres
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
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
Unconditional and Conditional Large Gaps between the zeros of the Riemann Zeta-Function
In this paper, first by employing inequalities derived from the Opial
inequality due to David Boyd with best constant, we will establish new
unconditional lower bounds for the gaps between the zeros of the Riemann zeta
function. Second, on the hypothesis that the moments of the Hardy Z-function
and its derivatives are correctly predicted, we establish some explicit
formulae for the lower bounds of the gaps between the zeros and use them to
establish some new conditional bounds. In particular it is proved that the
consecutive nontrivial zeros often differ by at least 6.1392 (conditionally)
times the average spacing. This value improves the value 4.71474396 that has
been derived in the literature
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