Continuous symmetrized Sobolev inner products of order \(N\). I (Q1779341)

Let \(\{ Q_n\}\) be a sequence of orthogonal polynomials satisfying \[ Q_{2n}(x)=P_n(x^2), \quad Q_{2n+1}(x)=x R_n(x^2) \] for sequences of polynomials \(P_n\) and \(R_n\). Chihara has studied how to identify the orthogonality conditions for \(P_n\)'s and \(R_n\)'s in the standard case. The authors extend this problem to the situation of Sobolev orthogonality with an arbitrary order of derivatives, and study the relation between the corresponding orthogonalities. The particular case of the so-called Freud-Sobolev orthogonal polynomials is analyzed.