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

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• 作者：Jonathan L. Gross^a ; ¹ ; ^{gross@cs.columbia.edu" class="auth_mail" title="E-mail the corresponding author} ; Toufik Mansour^b ; ^{tmansour@univ.haifa.ac.il" class="auth_mail" title="E-mail the corresponding author} ; Thomas W. Tucker^c ; ² ; ^{ttucker@colgate.edu" class="auth_mail" title="E-mail the corresponding author} ; David G.L. Wang^d ; ^e ; ³ ; ^{glw@bit.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Dirichlet's approximation theorem ; Real-rooted polynomial ; Recurrence ; Root geometry
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 September 2016
• 年：2016
• 卷：441
• 期：2
• 页码：499-528
• 全文大小：582 K

文摘

We consider the sequence of polynomials class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300713&_mathId=si1.gif&_user=111111111&_pii=S0022247X16300713&_rdoc=1&_issn=0022247X&md5=d49a21a0b0154ef53cf0f1f059fcc7ce" title="Click to view the MathML source">W_n(x)class="mathContainer hidden">class="mathCode">$W_n(x)$ defined by the recursion class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300713&_mathId=si2.gif&_user=111111111&_pii=S0022247X16300713&_rdoc=1&_issn=0022247X&md5=0a071600082e2921194e20beab907c4b" title="Click to view the MathML source">W_n(x)=(ax+b)W_n−1(x)+dW_n−2(x)class="mathContainer hidden">class="mathCode">$W_n(x)=(ax+b)W_{n-1}(x)+d{W}_{n-2}(x)$, with initial values class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300713&_mathId=si3.gif&_user=111111111&_pii=S0022247X16300713&_rdoc=1&_issn=0022247X&md5=a7c4d37d1b4bb018e3697d800cdfb396" title="Click to view the MathML source">W₀(x)=1class="mathContainer hidden">class="mathCode">$W_0(x)=1$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300713&_mathId=si4.gif&_user=111111111&_pii=S0022247X16300713&_rdoc=1&_issn=0022247X&md5=364822468baf92298d4232a1e2135272" title="Click to view the MathML source">W₁(x)=t(x−r)class="mathContainer hidden">class="mathCode">$W_1(x)=t(x-r)$, where a, b, d, t, r   are real numbers, with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300713&_mathId=si5.gif&_user=111111111&_pii=S0022247X16300713&_rdoc=1&_issn=0022247X&md5=245cf34d49df23b521ac410d80527c65" title="Click to view the MathML source">a,t>0class="mathContainer hidden">class="mathCode">$a,t>0$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300713&_mathId=si565.gif&_user=111111111&_pii=S0022247X16300713&_rdoc=1&_issn=0022247X&md5=9b21f69c8ffada0aab2cd02a7df5c15a" title="Click to view the MathML source">d<0class="mathContainer hidden">class="mathCode">$d<0$. It is known that every polynomial class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300713&_mathId=si1.gif&_user=111111111&_pii=S0022247X16300713&_rdoc=1&_issn=0022247X&md5=d49a21a0b0154ef53cf0f1f059fcc7ce" title="Click to view the MathML source">W_n(x)class="mathContainer hidden">class="mathCode">$W_n(x)$ is distinct-real-rooted. We find that, as class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300713&_mathId=si454.gif&_user=111111111&_pii=S0022247X16300713&_rdoc=1&_issn=0022247X&md5=bdee3e46e92bc54a5ea99a063745338b" title="Click to view the MathML source">n→∞class="mathContainer hidden">class="mathCode">$n\to \infty$, the smallest root of the polynomial class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300713&_mathId=si1.gif&_user=111111111&_pii=S0022247X16300713&_rdoc=1&_issn=0022247X&md5=d49a21a0b0154ef53cf0f1f059fcc7ce" title="Click to view the MathML source">W_n(x)class="mathContainer hidden">class="mathCode">$W_n(x)$ converges decreasingly to a real number, and that the largest root converges increasingly to a real number. Moreover, by using the Dirichlet approximation theorem, we prove that for every integer class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300713&_mathId=si47.gif&_user=111111111&_pii=S0022247X16300713&_rdoc=1&_issn=0022247X&md5=efe1b79435224c89924f2923fa2dac8c" title="Click to view the MathML source">j≥2class="mathContainer hidden">class="mathCode">$j\ge 2$, the j  th smallest root of the polynomial class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300713&_mathId=si1.gif&_user=111111111&_pii=S0022247X16300713&_rdoc=1&_issn=0022247X&md5=d49a21a0b0154ef53cf0f1f059fcc7ce" title="Click to view the MathML source">W_n(x)class="mathContainer hidden">class="mathCode">$W_n(x)$ converges as class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300713&_mathId=si454.gif&_user=111111111&_pii=S0022247X16300713&_rdoc=1&_issn=0022247X&md5=bdee3e46e92bc54a5ea99a063745338b" title="Click to view the MathML source">n→∞class="mathContainer hidden">class="mathCode">$n\to \infty$, and so does the jth largest root. It turns out that these two convergence points are independent of the numbers t, r, and the integer j. We obtain explicit expressions for the above four limit points.