Given a linear second-order differential operator L≡ϕD2+ψD\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {L}}\equiv \phi \,D^2+\psi \,D$$\end{document} with non zero polynomial coefficients of degree at most 2, a sequence of real numbers λn\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _n$$\end{document}, n⩾0\documentclass[12pt]{minimal}
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\begin{document}$$n\geqslant 0$$\end{document}, and a Sobolev bilinear form B(p,q)=∑k=0Nuk,p(k)q(k),N⩾0,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\mathcal {B}}(p,q)\,=\,\sum _{k=0}^N\left\langle {{\mathbf {u}}_k,\,p^{(k)}\,q^{(k)}}\right\rangle , \quad N\geqslant 0, \end{aligned}$$\end{document}where uk\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {u}}_k$$\end{document}, 0⩽k⩽N\documentclass[12pt]{minimal}
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\begin{document}$$0\leqslant k \leqslant N$$\end{document}, are linear functionals defined on polynomials, we study the orthogonality of the polynomial solutions of the differential equation L[y]=λny\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {L}}[y]=\lambda _n\,y$$\end{document} with respect to B\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {B}}$$\end{document}. We show that such polynomials are orthogonal with respect to B\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {B}}$$\end{document} if the Pearson equations D(ϕuk)=(ψ+kϕ′)uk\documentclass[12pt]{minimal}
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\begin{document}$$D(\phi \,{\mathbf {u}}_k)=(\psi +k\,\phi ')\,{\mathbf {u}}_k$$\end{document}, 0⩽k⩽N\documentclass[12pt]{minimal}
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\begin{document}$$0\leqslant k \leqslant N$$\end{document}, 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.
