Regular Sobolev type orthogonal polynomials: The Bessel case
We are dealing with a nice paper, studying Sobolev-type orthogonal polynomials arising from a regular linear functional on the linear space of polynomials with real coefficients in the following manner: \[ \varphi(p,q):=\langle u,pq\rangle+ \lambda p'(c)q'(c)\quad (\lambda, c\in\mathbb{R}). \] The ordinary multiplication operator is not selfadjoint for \(\varphi\), but luckily the multiplication operator \((x-c)^2\) is! Now using the functional \(u_1=(x-c)^2 u\), the authors derive Christoffel-Darboux formulae, 5-term recurrence relations and a recursive algorithm to calculate its coefficients for the orthogonal polynomials connected with \(u_1\). Finally, the general results are used to treat the linear functional \(u\) that gives rise to the Bessel polynomials, with the choice \(c=0\). The authors specialize their general results on this case, giving explicit formulae. Moreover, they give a Rodrigues type formula and the form of a second-order differential equation satisfied by the Sobolev orthogonal polynomials (instead of polynomial coefficients of degrees 2, 1 and 0, resp., the degrees are now 4, 3 and 2; no proof, however, is given that these coefficients do not have a common factor).
scientific article; zbMATH DE number 894660