Complex zero decreasing sequences

Abstract

Abstract. The purpose of this paper is to investigate the real sequences γ 0 , γ 1 , γ 2 , . . . with the property that if p(x) = n k=0 a k x k is any real polynomial, then n k=0 γ k a k x k has no more nonreal zeros than p(x). In particular, the authors establish a converse to a classical theorem of Laguerre.

On linear operators preserving the set of positive polynomials

“…An example illustrating (ii) is given in [2, p. 520] (see also [3,Example 1.8]). Namely, the sequence {1 + i + i 2 } ∞ i=0 corresponds to a diagonal transformation preserving hyperbolicity.…”

“…We will refer to such operators as diagonal transformations or diagonal sequences. Diagonal transformations preserving the set of positive polynomials are referred to as λ-sequences in the literature (see [6, p. 110], and [2,3,8]). Reserving the symbol Φ for general linear operators we use in this section the notation T ∈ Λ to indicate that T is a diagonal transformation preserving positivity.…”

“…First we want to recall what was previously known about positivity-and non-negativity-preservers in the classical case of linear operators acting diagonally in the standard monomial basis of R[x] and secondly we want to point out some (known to specialists [2], [3]) mistakes in Ch. 4 of the important treatise [6].…”

The distribution of zeros of a class of Jacobi polynomials

“…The polynomials g m (z) [22, p. 115], [9] or [7] [9]). The next theorem says that multiplier sequences are also stability-preserving operators.…”

A new property of a class of Jacobi polynomials

“…A solution of Problem 3 could have interesting ramifications in the theory of distribution of zeros of polynomials and transcendental entire functions in the Laguerre-Pólya class (see, for example, [4] or [5]). …”