Root geometry of polynomial sequences I: Type (0, 1)

Jonathan L. Gross; Toufik Mansour; Thomas W. Tucker; David G.L. Wang

doi:10.1016/j.jmaa.2015.08.039

Root geometry of polynomial sequences I: Type (0, 1)

Jonathan L. Gross, Toufik Mansour, Thomas W. Tucker, David G.L. Wang^*

^*Corresponding author for this work

School of Mathematics and Statistics

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

Abstract

This paper concerns the distribution in the complex plane of the roots of a polynomial sequence {Wn(x)}n≥0 given by a recursion W_n(x)=aW_n-1(x)+(bx+c)W_n-2(x), with W₀(x)=1 and W₁(x)=t(x-r), where a>0, b>0, and c, t, r∈R. Our results include proof of the distinct-real-rootedness of every such polynomial W_n(x), derivation of the best bound for the zero-set {x|W_n(x)=0for some n≥1}, and determination of three precise limit points of this zero-set. Also, we give several applications from combinatorics and topological graph theory.

Original language	English
Article number	19736
Pages (from-to)	1261-1289
Number of pages	29
Journal	Journal of Mathematical Analysis and Applications
Volume	433
Issue number	2
DOIs	https://doi.org/10.1016/j.jmaa.2015.08.039
Publication status	Published - 15 Jan 2016

Keywords

Genus distribution
Real-rooted polynomial
Recurrence
Root geometry

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10.1016/j.jmaa.2015.08.039

Cite this

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title = "Root geometry of polynomial sequences I: Type (0, 1)",

abstract = "This paper concerns the distribution in the complex plane of the roots of a polynomial sequence {Wn(x)}n≥0 given by a recursion Wn(x)=aWn-1(x)+(bx+c)Wn-2(x), with W0(x)=1 and W1(x)=t(x-r), where a>0, b>0, and c, t, r∈R. Our results include proof of the distinct-real-rootedness of every such polynomial Wn(x), derivation of the best bound for the zero-set {x|Wn(x)=0for some n≥1}, and determination of three precise limit points of this zero-set. Also, we give several applications from combinatorics and topological graph theory.",

keywords = "Genus distribution, Real-rooted polynomial, Recurrence, Root geometry",

author = "Gross, {Jonathan L.} and Toufik Mansour and Tucker, {Thomas W.} and Wang, {David G.L.}",

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doi = "10.1016/j.jmaa.2015.08.039",

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T2 - Type (0, 1)

AU - Gross, Jonathan L.

AU - Mansour, Toufik

AU - Tucker, Thomas W.

AU - Wang, David G.L.

PY - 2016/1/15

Y1 - 2016/1/15

N2 - This paper concerns the distribution in the complex plane of the roots of a polynomial sequence {Wn(x)}n≥0 given by a recursion Wn(x)=aWn-1(x)+(bx+c)Wn-2(x), with W0(x)=1 and W1(x)=t(x-r), where a>0, b>0, and c, t, r∈R. Our results include proof of the distinct-real-rootedness of every such polynomial Wn(x), derivation of the best bound for the zero-set {x|Wn(x)=0for some n≥1}, and determination of three precise limit points of this zero-set. Also, we give several applications from combinatorics and topological graph theory.

AB - This paper concerns the distribution in the complex plane of the roots of a polynomial sequence {Wn(x)}n≥0 given by a recursion Wn(x)=aWn-1(x)+(bx+c)Wn-2(x), with W0(x)=1 and W1(x)=t(x-r), where a>0, b>0, and c, t, r∈R. Our results include proof of the distinct-real-rootedness of every such polynomial Wn(x), derivation of the best bound for the zero-set {x|Wn(x)=0for some n≥1}, and determination of three precise limit points of this zero-set. Also, we give several applications from combinatorics and topological graph theory.

KW - Genus distribution

KW - Real-rooted polynomial

KW - Recurrence

KW - Root geometry

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M3 - Article

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EP - 1289

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 2

M1 - 19736

ER -

Root geometry of polynomial sequences I: Type (0, 1)

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Keywords

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