On Sobolev bilinear forms and polynomial solutions of second-order differential equations

Abstract

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

Sobolev orthogonality of polynomial solutions of second-order partial differential equations

“…(1,1) n−2 (y), for n 2 (see [Garcia-Ardila and Marriaga (2021),eq. (4.14)]), then for n 2 and 2 m n,…”

“…satisfy (6.7) and (6.8) (see Garcia-Ardila and Marriaga (2021) and the references therein), respectively. Then, in this case, the simplex polynomials are given by P (α,β,−k) n,m (x, y)…”

On Sobolev orthogonal polynomials on a triangle

“…Some references include studies on the unit ball and the unit sphere [3-5, 11, 12, 15, 16, 18, 19], the simplex [1,20], product domains [6,7,9] and other interesting domains [14]. We remark that some Sobolev orthogonal polynomials are eigenfunctions of second order linear ordinary differential equations (see, for instance, [10] and the references therein) and second order linear partial differential equations [2].…”