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• W1489090542 abstract "We revisit the ladder operators for orthogonal polynomials and re-interpret two supplementary conditions as compatibility conditions of two linear over-determined systems; one involves the variation of the polynomials with respect to the variable <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=z> <mml:semantics> <mml:mi>z</mml:mi> <mml:annotation encoding=application/x-tex>z</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (spectral parameter) and the other a recurrence relation in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (the lattice variable). For the Jacobi weight <disp-formula content-type=math/mathml> [ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=w left-parenthesis x right-parenthesis equals left-parenthesis 1 minus x right-parenthesis Superscript alpha Baseline left-parenthesis 1 plus x right-parenthesis Superscript beta Baseline comma x element-of left-bracket negative 1 comma 1 right-bracket comma> <mml:semantics> <mml:mrow> <mml:mi>w</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mi>x</mml:mi> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>α<!-- α --></mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>x</mml:mi> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>β<!-- β --></mml:mi> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:mspace width=2em /> <mml:mi>x</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mo stretchy=false>[</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>]</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>w(x)=(1-x)^{alpha }(1+x)^{beta },qquad xin [-1,1],</mml:annotation> </mml:semantics> </mml:math> ] </disp-formula> we show how to use the compatibility conditions to explicitly determine the recurrence coefficients of the monic Jacobi polynomials." @default.
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• W1489090542 date "2004-08-30" @default.
• W1489090542 modified "2023-10-02" @default.
• W1489090542 title "Jacobi polynomials from compatibility conditions" @default.
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• W1489090542 doi "https://doi.org/10.1090/s0002-9939-04-07566-5" @default.
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