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Everitt, W.N.,Kwon, K.H.,Littlejohn, L.L.,Wellman, R.,Yoon, G.J. Koninklijke Vlaamse Ingenieursvereniging 2007 Journal of computational and applied mathematics Vol.208 No.1
<P><B>Abstract</B></P><P>We develop the left-definite analysis associated with the self-adjoint Jacobi operator Ak(α,β), generated from the classical second-order Jacobi differential expression<SUB>ℓα,β,k</SUB>[y](t)=1<SUB>wα,β</SUB>(t)((-(1-t<SUP>)α+1</SUP>(1+t<SUP>)β+1</SUP><SUP>y′</SUP>(t)<SUP>)′</SUP>+k(1-t<SUP>)α</SUP>(1+t<SUP>)β</SUP>y(t))(t∈(-1,1)),in the Hilbert space Lα,β2(-1,1)≔<SUP>L2</SUP>((-1,1);<SUB>wα,β</SUB>(t)), where <SUB>wα,β</SUB>(t)=(1-t<SUP>)α</SUP>(1+t<SUP>)β</SUP>, that has the Jacobi polynomials {Pm(α,β)}m=0∞ as eigenfunctions; here, α,β>-1 and <I>k</I> is a fixed, non-negative constant. More specifically, for each n∈N, we explicitly determine the unique left-definite Hilbert–Sobolev space Wn,k(α,β)(-1,1) and the corresponding unique left-definite self-adjoint operator Bn,k(α,β) in Wn,k(α,β)(-1,1) associated with the pair (Lα,β2(-1,1),Ak(α,β)). The Jacobi polynomials {Pm(α,β)}m=0∞ form a complete orthogonal set in each left-definite space Wn,k(α,β)(-1,1) and are the eigenfunctions of each Bn,k(α,β). Moreover, in this paper, we explicitly determine the domain of each Bn,k(α,β) as well as each integral power of Ak(α,β). The key to determining these spaces and operators is in finding the explicit Lagrangian symmetric form of the integral composite powers of <SUB>ℓα,β,k</SUB>[·]. In turn, the key to determining these powers is a double sequence of numbers which we introduce in this paper as the <I>Jacobi–Stirling numbers</I>. Some properties of these numbers, which in some ways behave like the classical Stirling numbers of the second kind, are established including a remarkable, and yet somewhat mysterious, identity involving these numbers and the eigenvalues of Ak(α,β).</P>
Diagonalizability and symmetrizability of Sobolev-type bilinear forms: A combinatorial approach
Kim, H.K.,Kwon, K.H.,Littlejohn, L.L.,Yoon, G.J. North Holland [etc.] 2014 Linear algebra and its applications Vol.460 No.-
In an earlier paper, Kwon, Littlejohn and Yoon characterized symmetric Sobolev bilinear forms and showed that they have, like symmetric matrices, a diagonal representation. In this paper, we present a new proof of one of their main results by interpreting the coefficients in the diagonal representation of a Sobolev-type bilinear form from a combinatorial point of view. We view this as an improvement over the original proof which relied on mathematical induction.
Ghost matrices and a characterization of symmetric Sobolev bilinear forms
Kwon, K.H.,Littlejohn, Lance L.,Yoon, G.J. Elsevier 2009 Linear algebra and its applications Vol.431 No.1
<P><B>Abstract</B></P><P>In this paper, we characterize symmetric Sobolev bilinear forms defined on P×P, where P is the space of polynomials. More specifically we show that symmetric Sobolev bilinear forms, like symmetric matrices, can be re-written with a diagonal representation. As an application, we introduce the notion of a ghost matrix, extending some classic work of Stieltjes.</P>
Sobolev orthogonal polynomials in two variables and second order partial differential equations
Lee, Jeong Keun,Littlejohn, L.L. Elsevier 2006 Journal of mathematical analysis and applications Vol.322 No.2
<P><B>Abstract</B></P><P>We consider polynomials in two variables which satisfy an admissible second order partial differential equation of the form<ce:label>(∗)</ce:label>A<SUB>uxx</SUB>+2B<SUB>uxy</SUB>+C<SUB>uyy</SUB>+D<SUB>ux</SUB>+E<SUB>uy</SUB>=λu, and are orthogonal relative to a symmetric bilinear form defined byϕ(p,q)=〈σ,pq〉+〈τ,<SUB>px</SUB><SUB>qx</SUB>〉, where A,…,E are polynomials in <I>x</I> and <I>y</I>, <I>λ</I> is an eigenvalue parameter, <I>σ</I> and <I>τ</I> are linear functionals on polynomials. We find a condition for the partial differential equation <ce:cross-ref refid='fd001'>(∗)</ce:cross-ref> to have polynomial solutions which are orthogonal relative to a symmetric bilinear form ϕ(⋅,⋅). Also examples are provided.</P>
Construction of differential operators having Bochner–Krall orthogonal polynomials as eigenfunctions
Kwon, K.H.,Littlejohn, L.L.,Yoon, G.J. Elsevier 2006 Journal of mathematical analysis and applications Vol.324 No.1
<P><B>Abstract</B></P><P>Suppose {<SUB>Qn</SUB>}n=0∞ is a sequence of polynomials orthogonal with respect to the moment functional τ=σ+ν, where <I>σ</I> is a classical moment functional (Jacobi, Laguerre, Hermite) and <I>ν</I> is a point mass distribution with finite support. In this paper, we develop a new method for constructing a differential equation having {<SUB>Qn</SUB>}n=0∞ as eigenfunctions.</P>
ORTHOGONAL POLYNOMIALS SATISFYING PARTIAL DIFFERENTIAL EQUATIONS BELONGING TO THE BASIC CLASS
Lee, J.K.,L.L. Littlejohn,Yoo, B.H. Korean Mathematical Society 2004 대한수학회지 Vol.41 No.6
We classify all partial differential equations with polynomial coefficients in $\chi$ and y of the form A($\chi$) $u_{{\chi}{\chi}}$ + 2B($\chi$, y) $u_{{\chi}y}$ + C(y) $u_{yy}$ + D($\chi$) $u_{{\chi}}$ + E(y) $u_{y}$ = λu, which has weak orthogonal polynomials as solutions and show that partial derivatives of all orders are orthogonal. Also, we construct orthogonal polynomials in d-variables satisfying second order partial differential equations in d-variables.s.