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• W3201230558 abstract "Abstract Given a linear second-order differential operator $${mathcal {L}}equiv phi ,D^2+psi ,D$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>≡</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mspace /> <mml:msup> <mml:mi>D</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>ψ</mml:mi> <mml:mspace /> <mml:mi>D</mml:mi> </mml:mrow> </mml:math> with non zero polynomial coefficients of degree at most 2, a sequence of real numbers $$lambda _n$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> , $$ngeqslant 0$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , and a Sobolev bilinear form $$begin{aligned} {mathcal {B}}(p,q),=,sum _{k=0}^Nleftlangle {{mathbf {u}}_k,,p^{(k)},q^{(k)}}rightrangle , quad Ngeqslant 0, end{aligned}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mi>B</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mspace /> <mml:mo>=</mml:mo> <mml:mspace /> <mml:munderover> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mi>N</mml:mi> </mml:munderover> <mml:mfenced> <mml:mrow> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:mspace /> <mml:msup> <mml:mi>p</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> <mml:mspace /> <mml:msup> <mml:mi>q</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:mfenced> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mi>N</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> where $${mathbf {u}}_k$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> , $$0leqslant k leqslant N$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>⩽</mml:mo> <mml:mi>k</mml:mi> <mml:mo>⩽</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> , are linear functionals defined on polynomials, we study the orthogonality of the polynomial solutions of the differential equation $${mathcal {L}}[y]=lambda _n,y$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>L</mml:mi> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>y</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mspace /> <mml:mi>y</mml:mi> </mml:mrow> </mml:math> with respect to $${mathcal {B}}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>B</mml:mi> </mml:math> . We show that such polynomials are orthogonal with respect to $${mathcal {B}}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>B</mml:mi> </mml:math> if the Pearson equations $$D(phi ,{mathbf {u}}_k)=(psi +k,phi '),{mathbf {u}}_k$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>D</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mspace /> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ψ</mml:mi> <mml:mo>+</mml:mo> <mml:mi>k</mml:mi> <mml:mspace /> <mml:msup> <mml:mi>ϕ</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mspace /> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> </mml:math> , $$0leqslant k leqslant N$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>⩽</mml:mo> <mml:mi>k</mml:mi> <mml:mo>⩽</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> , are satisfied by the linear functionals in the bilinear form. Moreover, we use our results as a general method to deduce the Sobolev orthogonality for polynomial solutions of differential equations associated with classical orthogonal polynomials with negative integer parameters." @default.
• W3201230558 created "2021-09-27" @default.
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• W3201230558 date "2021-09-17" @default.
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• W3201230558 title "On Sobolev bilinear forms and polynomial solutions of second-order differential equations" @default.
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