On the Sobolev orthogonality of classical orthogonal polynomials for non standard parameters

Abstract

The discrete part of the discrete-continuous orthogonality \[ \mathscr {B}(f,g)=\mathscr {B}_d( f,g)+\mathscr {B}_c(f^{(N)},g^{(N)}), \] is studied for families of classical orthogonal polynomials such that the associated three-term recurrence relation \[ xp_n=p_{n+1}+\beta _np_n+ \gamma _n p_{n-1}, \] presents one vanishing coefficient $\gamma _n$, as in the case of Laguerre polynomials $L_n^{(-N)}$, Jacobi polynomials $P_n^{(-N,\beta )}$ and Gegenbauer polynomials $C_n^{(-N+1/2)}$ with $N\in \mathbb {N}$. It is shown that the discrete bilinear functional $\mathscr {B}_d$ can be replaced by a linear functional, $\mathscr {L}$, or by another bilinear functional related with $\mathscr {L}$, which allows us to reformulate the orthogonality in a much simpler way in the case of Laguerre polynomials and in a totally explicit form in the case of Jacobi and Gegenbauer polynomials.