Geodesic
Weakly Increasing Zero-Diminishing Sequences
Serdica Mathematical Journal, Tome 22 (1996) no. 4, pp. 547-570.

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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.
Keywords: Weakly Increasing Sequences, Zero-Diminishing Sequences, Zeros of Entire Functions, Interpolation 
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     title = {Weakly {Increasing} {Zero-Diminishing} {Sequences}},
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Bakan, Andrew; Craven, Thomas; Csordas, George; Golub, Anatoly. Weakly Increasing Zero-Diminishing Sequences. Serdica Mathematical Journal, Tome 22 (1996) no. 4, pp. 547-570. http://geodesic.mathdoc.fr/item/SMJ2_1996_22_4_a4/