Generalized Gegenbauer orthogonal polynomials

In this paper we explore a specific semi-classical orthogonal sequence, namely the generalized Gegenbauer orthogonal polynomials (GG) which appear in many applications such as the weighted Lp mean convergence of Hermite–Fejér interpolation or the chain of harmonic oscillators in the absence of externally applied forces. First we trace back the genesis of GG underlining its links with the Jacobi orthogonal polynomials. Second we establish a differential–difference relation and the second-order differential equation satisfied by this sequence. We end by giving the fourth-order differential equation satisfied by the association (of arbitrary order) of the GG.