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Sobolev-type orthogonal polynomials and their zeros. (English) Zbl 0891.33005
The authors study the general symmetric bilinear form \[ \phi (p,q):=\langle \sigma,pq\rangle+\lambda p^{(r)}(a)q^{(r)}(a) + \mu p^{(s)}(b)q^{(s)}(b), \] where \(\sigma\) is a quasi definite moment functional on \({\mathcal P}=\mathbf{C}[x]\) (the linear space of all polynomials); \(\mu,a,b\in\)C, \(\lambda\in\mathbf{C}\setminus\{0\}\) and \(r,s\) non-negative integers with \(0\leq r<s\) (\(r<s\) in case \(a=b\)).
They derive results on the following subjects:
(1) a necessary and sufficient condition for \(\phi (\cdot,\cdot)\) to be quasi definite (equivalent with the existence of a unique monic orthogonal polynomial sequence (MOPS) with respect to the bilinear form),
(2) a recurrence relation for the MOPS (length depending on the values of \(\mu,r,s\) and whether \(a=b\) or \(a\not= b\)),
(3) a reproducing kernel property,
(4) a Christoffel-Darboux formula,
(5) quasi-orthogonality with respect to \(\sigma\),
(6) a second order differential equation with polynomial coefficients of bounded degree,
(7) zeros of the MOPS in comparison with the location of those for the orthogonal polynomials connected with \(\sigma\).
The general treatment of the subject and the derivation of results is quite elegant; it is possible to derive many results on special cases, scattered through the literature, along with new results.

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)