Krein–Sobolev Orthogonal Polynomials II

In a recent paper, Littlejohn and Quintero studied the orthogonal polynomials {Kn}n=0∞—which they named Krein–Sobolev polynomials—that are orthogonal in the classical Sobolev space H1[−1,1] with respect to the (positive-definite) inner product (f,g)1,c:=−f(1)−f(−1)g¯(1)−g¯(−1)2+∫−11(f′(x)g¯′(x)+cf(x)g¯(x))dx, 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 Kc(c>0) in L2(−1,1). Other than K0 and K1, these polynomials are not eigenfunctions of Kc. As shown by Littlejohn and Quintero, the sequence {Kn}n=0∞ forms a complete orthogonal set in the first left-definite space (H1[−1,1],(·,·)1,c) associated with (Kc,L2(−1,1)). Furthermore, they show that, for n≥1,Kn(x) has n distinct zeros in (−1,1). In this note, we find an explicit formula for Krein–Sobolev polynomials {Kn}n=0∞.