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

In this paper, we develop the left-definite spectral theory associated with the self-adjoint operator A(k) in L2(- 1, 1), generated from the classical second-order Legendre differential equation lL,k[y](t) = -((1 - t2)y')' + ky = λy (t∈ (-1,1)), that has the Legendre polynomials {Pm(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 Wn(k) and its associated inner product (.,.)n,k for each n ∈ N. Moreover, for each n ∈ N, we determine the corresponding unique left-definite self-adjoint operator An(k) in Wn(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 lL,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.