Orthogonal polynomials relative to linear perturbations of quasi-definite moment functionals

Bull. Korean Math. Soc. 1999 Vol. 36, No. 3, 543-564

K. H. Kwon, D. W. Lee, and J. H. Lee KAIST, Kyungpook National University, KAIST

Abstract : Consider a symmetric bilinear form defined on $\Pi\times \Pi$ by \begin{equation*} \la f ,g \ra_{\lambda,\mu} = \la \si , f g \ra + \lambda L[f](a) L[g](a) + \mu M[f](b) M[g](b) , \end{equation*} where $\si$ is a quasi-definite moment functional, $L$ and $M$ are linear operators on $\Pi$, the space of all real polynomials and $a,b,\lambda$, and $\mu$ are real constants. We find a necessary and sufficient condition for the above bilinear form to be quasi-definite and study various properties of corresponding orthogonal polynomials. This unifies many previous works which treated cases when both $L$ and $M$ are differential or difference operators. Finally, infinite order operator equations having such orthogonal polynomials as eigenfunctions are given when $\mu = 0$.

Keywords : orthogonal polynomials, linear perturbation, quasi-definiteness