The orthogonal polynomials for the (continuous) Sobolev inner product \[ (f,g)= \int_{-1}^ 1 f(x) g(x) (1- x^ 2)^{\alpha- 1/2} dx+\lambda \int_{-1}^ 1 f'(x) g'(x) (1-x^ 2)^{\alpha- 1/2} dx \] are investigated. These Gegenbauer-Sobolev orthogonal polynomials are expressed in terms of the Gegenbauer polynomials by using an appropriate selfadjoint differential operator. This operator also gives a differential-difference relation and a Rodrigues-type formula. Finally it is shown that the Gegenbauer-Sobolev polynomials of degree \(2n\) have exactly \(2n\) real and distinct zeros, at most two zeros can be outside \([-1,1]\) and for \(\alpha\geq 1/2\) they are all in \([-1,1]\). Furthermore the zeros interlace with the zeros of the Gegenbauer polynomial of the same degree.
For the entire collection see [Zbl 0798.00017].