# Weakly Increasing Zero-Diminishing Sequences

• Volume: 22, Issue: 4, page 547-570
• ISSN: 1310-6600

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## Abstract

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The following problem, suggested by Laguerre’s Theorem (1884), remains open: Characterize all real sequences {μk} k=0...∞ which have the zero-diminishing property; that is, if k=0...n, p(x) = ∑(ak x^k) is any P real polynomial, then k=0...n, p(x) = ∑(μk ak x^k) has no more real zeros than p(x). In this paper this problem is solved under the additional assumption of a weak growth condition on the sequence {μk} k=0...∞, namely lim n→∞ | μn |^(1/n) < ∞. More precisely, it is established that the real sequence {μk} k≥0 is a weakly increasing zerodiminishing sequence if and only if there exists σ ∈ {+1,−1} and an entire function n≥1, Φ(z)= be^(az) ∏(1+ x/αn), a, b ∈ R^1, b =0, αn > 0 ∀n ≥ 1, ∑(1/αn) < ∞, such that µk = (σ^k)/Φ(k), ∀k ≥ 0.

## How to cite

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Bakan, Andrew, et al. "Weakly Increasing Zero-Diminishing Sequences." Serdica Mathematical Journal 22.4 (1996): 547-570. <http://eudml.org/doc/11651>.

@article{Bakan1996,
abstract = {The following problem, suggested by Laguerre’s Theorem (1884), remains open: Characterize all real sequences \{μk\} k=0...∞ which have the zero-diminishing property; that is, if k=0...n, p(x) = ∑(ak x^k) is any P real polynomial, then k=0...n, p(x) = ∑(μk ak x^k) has no more real zeros than p(x). In this paper this problem is solved under the additional assumption of a weak growth condition on the sequence \{μk\} k=0...∞, namely lim n→∞ | μn |^(1/n) < ∞. More precisely, it is established that the real sequence \{μk\} k≥0 is a weakly increasing zerodiminishing sequence if and only if there exists σ ∈ \{+1,−1\} and an entire function n≥1, Φ(z)= be^(az) ∏(1+ x/αn), a, b ∈ R^1, b =0, αn > 0 ∀n ≥ 1, ∑(1/αn) < ∞, such that µk = (σ^k)/Φ(k), ∀k ≥ 0.},
author = {Bakan, Andrew, Craven, Thomas, Csordas, George, Golub, Anatoly},
journal = {Serdica Mathematical Journal},
keywords = {Weakly Increasing Sequences; Zero-Diminishing Sequences; Zeros of Entire Functions; Interpolation; weakly increasing sequences; zero-diminishing sequences; zeros of entire functions; interpolation},
language = {eng},
number = {4},
pages = {547-570},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {Weakly Increasing Zero-Diminishing Sequences},
url = {http://eudml.org/doc/11651},
volume = {22},
year = {1996},
}

TY - JOUR
AU - Bakan, Andrew
AU - Craven, Thomas
AU - Csordas, George
AU - Golub, Anatoly
TI - Weakly Increasing Zero-Diminishing Sequences
JO - Serdica Mathematical Journal
PY - 1996
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 22
IS - 4
SP - 547
EP - 570
AB - The following problem, suggested by Laguerre’s Theorem (1884), remains open: Characterize all real sequences {μk} k=0...∞ which have the zero-diminishing property; that is, if k=0...n, p(x) = ∑(ak x^k) is any P real polynomial, then k=0...n, p(x) = ∑(μk ak x^k) has no more real zeros than p(x). In this paper this problem is solved under the additional assumption of a weak growth condition on the sequence {μk} k=0...∞, namely lim n→∞ | μn |^(1/n) < ∞. More precisely, it is established that the real sequence {μk} k≥0 is a weakly increasing zerodiminishing sequence if and only if there exists σ ∈ {+1,−1} and an entire function n≥1, Φ(z)= be^(az) ∏(1+ x/αn), a, b ∈ R^1, b =0, αn > 0 ∀n ≥ 1, ∑(1/αn) < ∞, such that µk = (σ^k)/Φ(k), ∀k ≥ 0.
LA - eng
KW - Weakly Increasing Sequences; Zero-Diminishing Sequences; Zeros of Entire Functions; Interpolation; weakly increasing sequences; zero-diminishing sequences; zeros of entire functions; interpolation
UR - http://eudml.org/doc/11651
ER -

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