Title: Legendre polynomials and Legendre-Stirling Numbers
Speaker: Lance Littlejohn, Baylor University
Seminary Type: Analysis and Operator Theory
Abstract: In this lecture, we will discuss properties of the \(n^{th}\) composite power of the self-adjoint operator \(T\) in \(L^{2}(-1,1)\), generated by the classical second-order Legendre differential expression \[ \ell\lbrack y](x)=-\left( (1-x^{2})y^{\prime}(x)\right) ^{\prime}% +ky(x)\quad(k\geq0\text{ fixed constant};\text{ }x\in(-1,1)), \] which has the Legendre polynomials \(\{P_{m}\}_{m=0}^{\infty}\) as eigenfunctions. These powers involve a sequence of numbers which we call the \textit{Legendre-Stirling numbers}. The \(n^{th}\) powers arise from a left-definite spectral study of the Legendre expression. These combinatorial numbers mimic in many ways the classical Stirling numbers of the second kind and we will discuss various properties of these numbers, including recent combinatorial interpretations and asymptotic properties of these numbers. Time permitting, extensions of these results to the general Jacobi case will also be mentioned.
