A Non-Standard Class of Sobolev Orthogonal Polynomials
Commun. Korean Math. Soc. 1997 Vol. 12, No. 4, 935-950
S. S. Han, I. H. Jung, K. H. Kwon, J. K. Lee Myongji university, KAIST, KAIST, Sunmoon university
Abstract : When $\tau$ is a quasi-definite moment functional on $\Cal P$, the vector space of all real polynomials, we consider a symmetric bilinear form $\phi(\cdot,\cdot)$ on $\Cal P \times \Cal P$ defined by $$ \phi(p,q) = \lambda p(a)q(a) + \mu p(b)q(b) + \la \tau , p' q' \ra , $$where $\lambda, \mu, a,$ and $b$ are real numbers. We first find a necessary and sufficient condition for $\phi(\cdot,\cdot)$ to be quasi-definite. When $\tau$ is a semi-classical moment functional, we discuss algebraic properties of the orthogonal polynomials relative to $\phi(\cdot,\cdot)$ and show that such orthogonal polynomials satisfy a fifth order differential equation with polynomial coefficients.