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 L-2(-1,1), generated from the classical second-order Legendre differential equation lL,k[y](t) = -((1 - t(2))y')' + ky = lambday (t is an element of (-1, 1)), that has the Legendre polynomials {P-m(t)}(m=0)(infinity) 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,(k) and its associated inner product ((.),(.))(n,k) for each n is an element of N. Moreover, for each n is an element of N, we determine the corresponding unique left-definite self-adjoint operator A,(k) in W,(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. (C) 2002 Elsevier Science B.V. All rights reserved.