Legendre polynomials, Legendre--Stirling numbers, and the left-definite spectral analysis of the Legendre differential expression (Q1860442)

The authors apply left-definite theory to the self-adjoint Legendre differential operator \(A(K)\), generated by the classical second order Legendre differential equation \(\ell_{L,K}[y](t)= -(1-t^2)y''+ 2ty'+ky=\lambda y\), \(t\in (-1,1)\), \(K>0\) with eigenfunctions as Legendre polynomials. The right-definite setting in this case is the Hilbert space \(H=L^2(-1,1)\). They determine the corresponding unique left-definite self-adjoint operator \(A_n(K)\) and characterize its domain in terms of another left-definite space. The key to determining these spaces and inner products is in finding the explicit Lagrangian symmetric form of the integral composite powers of \(\ell_{L,K} [\cdot]\). The key to determining these powers is a new identity involving a double sequence of numbers which they call Legendre-Stirling numbers.