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

摘要

In this paper, we develop the left-definite spectral theory associated with the self-adjoint operator A(k) in L~2(-1,1), generated from the classical second-order Legendre differential equation l_(L,k)[y](t) = -((1-t~2)y')' + ky = λy (t ∈ (-1,1)), that has the Legendre polynomials {P(m(t)}_(m = 0)~∞ as eigenfunctions; here, k is a fixed, nonnegative constant. More specifically, for k > 0, we explicitly determine the unique left-definite Hilbert-Sobolev space W_n(k) and its associated inner product(.,.)_(n,k) for each n ∈N, we determine the corresponding unique left-definite self-adjoint operator A_n(k) in W_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 l_(L,k)[·], In turn, the key to determining these powers is a remarkable new identity involving a double sequence of numbers which we call Legendre-Stirling numbers.