We consider the following discrete Sobolev inner product involving the Gegenbauer weight $$(f,g)_S:=\int_{ -1}^1f(x)g(x)(1-x^2)^{ \alpha}dx+M\big[f^{ (j)}(-1)g^{ (j)}(-1)+f^{ (j)}(1)g^{ (j)}(1)\big],$$ where $\alpha>-1,$ $j\in \mathbb{ N}\cup \{ 0\},$ and $M>0.$ Let $\{ Q_n^{ (\alpha,M,j)}\}_{ n\geq0}$ be the sequence of orthogonal polynomials with respect to the above inner product. These polynomials are eigenfunctions of a differential operator $\mathbf{ T}. $ We establish the asymptotic behavior of the corresponding eigenvalues. Furthermore, we calculate the exact value $$r_0 = \lim_{ n\rightarrow \infty}\frac{ \log \left(\max_{ x\in [-1,1]} |\widetilde{ Q}_n^{ (\alpha,M,j)}(x)|\right)}{ \log \widetilde{ \lambda}_n},$$ where $\{ \widetilde{ Q}_n^{ (\alpha,M,j)}\}_{ n\geq0}$ are the sequence of orthonormal polynomials with respect to this Sobolev inner product. This value $r_0$ is related to the convergence of a series in a left--definite space. Finally, we study the Mehler--Heine type asymptotics for $\{ Q_n^{ (\alpha,M,j)}\}_{ n\geq0}.$
Categories: math.CA