2004/08/30 by Yang Chen, Mourad E. H. Ismail, Mourad Ismail · 7 citations
Mathematics · #Random Matrices and Applications #Algebraic structures and combinatorial models #Advanced Combinatorial Mathematics
paper · pdf · doi:10.1090/s0002-9939-04-07566-5
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 </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 </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)α (1+x)β , x∈ [-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.