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

Abstract

We consider the following discrete Sobolev inner product involving the Gegenbauer weight $$(f,g{)}_S:={\int }_{-1}^{1}f(x)g(x)(1-x^2{)}^{\alpha }dx+M[f^{(j)}(-1)g^{(j)}(-1)+f^{(j)}(1)g^{(j)}(1)],$$ where $\alpha >-1,$ $j\in \mathbb{N}\cup \{0\},$ and $M>0.$ Let $\{Q_n^{(\alpha ,M,j)}{\}}_{n\ge 0}$ 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 $$r_0=\underset{n\to \infty }{\lim }\frac{\log \left({\max }_{x\in [-1,1]}|{\tilde{Q}}_n^{(\alpha ,M,j)}(x)|\right)}{\log {\tilde{\lambda }}_n},$$ where $\{{\tilde{Q}}_n^{(\alpha ,M,j)}{\}}_{n\ge 0}$ 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 $\{Q_n^{(\alpha ,M,j)}{\}}_{n\ge 0}.$