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 equationℓL,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.