Differential operator for discrete Gegenbauer--Sobolev orthogonal polynomials: eigenvalues and asymptotics

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 $\{Q_n^{(\alpha,M,j)}\}_{n\geq0}.$