26 research outputs found
On the asymptotic normality of the Legendre-Stirling numbers of the second kind
For the Legendre-Stirling numbers of the second kind asymptotic formulae are
derived in terms of a local central limit theorem. Thereby, supplements of the
recently published asymptotic analysis of the Chebyshev-Stirling numbers are
established. Moreover, we provide results on the asymptotic normality and
unimodality for modified Legendre-Stirling numbers
The Jacobi operator on and its various -functions
We offer a detailed treatment of spectral and Weyl-Titchmarsh-Kodaira theory
for all self-adjoint Jacobi operator realizations of the differential
expression \begin{align*} \tau_{\alpha,\beta} = - (1-x)^{-\alpha}
(1+x)^{-\beta}(d/dx) \big((1-x)^{\alpha+1}(1+x)^{\beta+1}\big) (d/dx),& \\
\alpha, \beta \in \mathbb{R}, \; x \in (-1,1),& \end{align*} in
, . In addition to discussing the separated boundary conditions that
lead to Jacobi orthogonal polynomials as eigenfunctions in detail, we
exhaustively treat the case of coupled boundary conditions and illustrate the
latter with the help of the general -periodic and Krein--von Neumann
extensions. In particular, we treat all underlying Weyl-Titchmarsh-Kodaira and
Green's function induced -functions and revisit their Nevanlinna-Herglotz
property. We also consider connections to other differential operators
associated with orthogonal polynomials such as Laguerre, Gegenbauer, and
Chebyshev.Comment: 59 pages. arXiv admin note: text overlap with arXiv:2102.00685,
arXiv:2110.15913, arXiv:1910.1311