FRINK Linked Data Fragments server

Matches in SemOpenAlex for { <https://semopenalex.org/work/W2964054920> ?p ?o ?g. }

• W2964054920 endingPage "5331" @default.
• W2964054920 startingPage "5323" @default.
• W2964054920 abstract "We study the differential equation <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=minus left-parenthesis p left-parenthesis x right-parenthesis y Superscript prime Baseline right-parenthesis Superscript prime Baseline plus q left-parenthesis x right-parenthesis y Superscript prime Baseline equals lamda y comma> <mml:semantics> <mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:msup> <mml:mi>y</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:msup> <mml:mi>y</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>- (p(x) y’)’ + q(x) y’ = lambda y,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>p(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a polynomial of degree at most 2 and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>q(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a polynomial of degree at most 1. This includes the classical Jacobi polynomials, Hermite polynomials, Legendre polynomials, Chebychev polynomials, and Laguerre polynomials. We provide a general electrostatic interpretation of zeros of such polynomials: a set of distinct, real numbers <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartSet x 1 comma ellipsis comma x Subscript n Baseline EndSet> <mml:semantics> <mml:mrow> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>}</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>left {x_1, dots , x_nright }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfies <disp-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p left-parenthesis x Subscript i Baseline right-parenthesis sigma-summation Underscript StartLayout 1st Row k equals 1 2nd Row k not-equals i EndLayout Overscript n Endscripts StartFraction 2 Over x Subscript k Baseline minus x Subscript i Baseline EndFraction equals q left-parenthesis x Subscript i Baseline right-parenthesis minus p prime left-parenthesis x Subscript i Baseline right-parenthesis normal f normal o normal r normal a normal l normal l 1 less-than-or-equal-to i less-than-or-equal-to n> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:munderover> <mml:mo>∑<!-- ∑ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mfrac linethickness=0> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≠<!-- ≠ --></mml:mo> <mml:mi>i</mml:mi> </mml:mrow> </mml:mfrac> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> </mml:mrow> </mml:munderover> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mfrac> <mml:mn>2</mml:mn> <mml:mrow> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>−<!-- − --></mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:mfrac> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:msup> <mml:mi>p</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:mspace width=2em /> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>f</mml:mi> <mml:mi mathvariant=normal>o</mml:mi> <mml:mi mathvariant=normal>r</mml:mi> <mml:mi mathvariant=normal>a</mml:mi> <mml:mi mathvariant=normal>l</mml:mi> <mml:mi mathvariant=normal>l</mml:mi> </mml:mrow> <mml:mtext> </mml:mtext> <mml:mn>1</mml:mn> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>i</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>begin{equation*} p(x_i) sum _{k = 1 atop k neq i}^{n}{frac {2}{x_k - x_i}} = q(x_i) - p’(x_i) qquad mathrm {for all}~ 1leq i leq n end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> if and only if they are zeros of a polynomial solving the differential equation. We also derive a system of ODEs depending on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p left-parenthesis x right-parenthesis comma q left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>p(x),q(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose solutions converge to the zeros of the orthogonal polynomial at an exponential rate." @default.
• W2964054920 created "2019-07-30" @default.
• W2964054920 creator A5054778975 @default.
• W2964054920 date "2018-09-17" @default.
• W2964054920 modified "2023-10-18" @default.
• W2964054920 title "Electrostatic interpretation of zeros of orthogonal polynomials" @default.
• W2964054920 cites W106526473 @default.
• W2964054920 cites W1596423227 @default.
• W2964054920 cites W1972794892 @default.
• W2964054920 cites W1994084361 @default.
• W2964054920 cites W1998336257 @default.
• W2964054920 cites W2004368850 @default.
• W2964054920 cites W2012187381 @default.
• W2964054920 cites W2019071695 @default.
• W2964054920 cites W2109817274 @default.
• W2964054920 cites W2120074357 @default.
• W2964054920 cites W2182758386 @default.
• W2964054920 cites W2183407769 @default.
• W2964054920 cites W2323702230 @default.
• W2964054920 cites W2334347857 @default.
• W2964054920 cites W2980671223 @default.
• W2964054920 doi "https://doi.org/10.1090/proc/14226" @default.
• W2964054920 hasPublicationYear "2018" @default.
• W2964054920 type Work @default.
• W2964054920 sameAs 2964054920 @default.
• W2964054920 citedByCount "10" @default.
• W2964054920 countsByYear W29640549202018 @default.
• W2964054920 countsByYear W29640549202019 @default.
• W2964054920 countsByYear W29640549202021 @default.
• W2964054920 countsByYear W29640549202022 @default.
• W2964054920 crossrefType "journal-article" @default.
• W2964054920 hasAuthorship W2964054920A5054778975 @default.
• W2964054920 hasBestOaLocation W29640549201 @default.
• W2964054920 hasConcept C11413529 @default.
• W2964054920 hasConcept C138885662 @default.
• W2964054920 hasConcept C154945302 @default.
• W2964054920 hasConcept C2776542307 @default.
• W2964054920 hasConcept C41008148 @default.
• W2964054920 hasConcept C41895202 @default.
• W2964054920 hasConceptScore W2964054920C11413529 @default.
• W2964054920 hasConceptScore W2964054920C138885662 @default.
• W2964054920 hasConceptScore W2964054920C154945302 @default.
• W2964054920 hasConceptScore W2964054920C2776542307 @default.
• W2964054920 hasConceptScore W2964054920C41008148 @default.
• W2964054920 hasConceptScore W2964054920C41895202 @default.
• W2964054920 hasIssue "12" @default.
• W2964054920 hasLocation W29640549201 @default.
• W2964054920 hasLocation W29640549202 @default.
• W2964054920 hasOpenAccess W2964054920 @default.
• W2964054920 hasPrimaryLocation W29640549201 @default.
• W2964054920 hasRelatedWork W2169996260 @default.
• W2964054920 hasRelatedWork W2357322933 @default.
• W2964054920 hasRelatedWork W2357839234 @default.
• W2964054920 hasRelatedWork W2364713684 @default.
• W2964054920 hasRelatedWork W2366383148 @default.
• W2964054920 hasRelatedWork W2376250058 @default.
• W2964054920 hasRelatedWork W2384022429 @default.
• W2964054920 hasRelatedWork W2389001606 @default.
• W2964054920 hasRelatedWork W4206632963 @default.
• W2964054920 hasRelatedWork W4248689280 @default.
• W2964054920 hasVolume "146" @default.
• W2964054920 isParatext "false" @default.
• W2964054920 isRetracted "false" @default.
• W2964054920 magId "2964054920" @default.
• W2964054920 workType "article" @default.