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We consider the following discrete Sobolev inner product involving theGegenbauer 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 polynomialswith respect to the above inner product. These polynomials are eigenfunctionsof a differential operator $\mathbf{T}. $ We establish the asymptotic behaviorof 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 oforthonormal 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$\{Q_n^{(\alpha,M,j)}\}_{n\geq0}.$

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