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Title: Weakly Increasing Zero-Diminishing Sequences
Authors: Bakan, Andrew
Craven, Thomas
Csordas, George
Golub, Anatoly
Keywords: Weakly Increasing Sequences
Zero-Diminishing Sequences
Zeros of Entire Functions
Issue Date: 1996
Publisher: Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Citation: Serdica Mathematical Journal, Vol. 22, No 4, (1996), 547p-570p
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.
ISSN: 1310-6600
Appears in Collections:Volume 22 Number 4

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