Krein–Sobolev Orthogonal Polynomials II - Alexander Jones; Lance Littlejohn; Alejandro Quintero Roba (Axioms)
In a recent paper, Littlejohn and Quintero studied the orthogonal polynomials <semantics><msubsup>Knn=0∞</msubsup></semantics>—which they named Krein–Sobolev polynomials—that are orthogonal in the classical Sobolev space <semantics>H1[−1,1]</semantics> with respect to the (positive-definite) inner product <semantics>(f,g)1,c:=−<mstyle scriptlevel="0" displaystyle="true"><mfenced separators="" open="(" close=")">f(1)−f(−1)</mfenced><mfenced separators="" open="(" close=")"><mover>g¯</mover>(1)−<mover>g¯</mover>(−1)</mfenced>2</mstyle>+<msubsup>∫−11</msubsup>(f′(x)<mover>g¯</mover>′(x)+cf(x)<mover>g¯</mover>(x))dx,</semantics> where c is a fixed, positive constant. These polynomials generalize the Althammer (or Legendre–Sobolev) polynomials first studied by Althammer and Schäfke. The Krein–Sobolev polynomials were found as a result of a left-definite spectral study of the self-adjoint Krein Laplacian operator <semantics>Kc</semantics><semantics>(c>0)</semantics> in <semantics>L2(−1,1).</semantics> Other than <semantics>K0</semantics> and <semantics>K1,</semantics> these polynomials are not eigenfunctions of <semantics>Kc.</semantics> As shown by Littlejohn and Quintero, the sequence <semantics><msubsup>Knn=0∞</msubsup></semantics> forms a complete orthogonal set in the first left-definite space <semantics>(H1[−1,1],(·,·)1,c)</semantics> associated with <semantics>(Kc,</semantics><semantics>L2(−1,1))</semantics>. Furthermore, they show that, for <semantics>n≥1,</semantics><semantics>Kn(x)</semantics> has n distinct zeros in <semantics>(−1,1).</semantics> In this note, we find an explicit formula for Krein–Sobolev polynomials <semantics><msubsup>Knn=0∞</msubsup></semantics>.